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STUDY OF FUNCTIONAL IDENTITIES

WITH DERIVATIONS IN NEAR RINGS

THESIS

SUBMITTED FOR THE AWARD OF THE DEGREE OF

Doctor of PhilosophyDoctor of PhilosophyDoctor of PhilosophyDoctor of Philosophy IN

MATHEMATICS

BY

FARHAT ALI

UNDER THE SUPERVISION OF

PROFESSOR ASMA ALI

DEPARTMENT OF MATHEMATICS

ALIGARH MUSLIM UNIVERSITY

ALIGARH (INDIA)

2016

DedicatedDedicatedDedicatedDedicated

ToToToTo

My BelovedMy BelovedMy BelovedMy Beloved

ParentsParentsParentsParents

April 2016

Farhat Ali Aligarh Muslim University, Aligarh

STUDY OF FUNCTIONAL IDENTITIES WITH DERIVATIONS IN NEAR RINGS

Contents

Acknowledgement (i-ii)

Preface (iii-vii)

Chapter 1 Preliminaries (1-18)

1.1 Introduction

1.2 Some near ring theoretic concepts

1.3 Some key results

Chapter 2 Functional identities with generalized semiderivations (19-38)

2.1 Introduction

2.2 Some preliminary definitions and results

2.3 Extension of Bell and Martindale’s results

2.4 Commutativity of prime near rings satisfying condition [F (U), F (U)] = {0}

Chapter 3 Functional identities with generalized semiderivations acting as

homomorphism and antihomomorphism (39-56)

3.1 Introduction

3.2 Results involving generalized semiderivations acting as homomorphism and

antihomomorphism

3.3 Some commutativity conditions involving generalized semiderivations

Chapter 4 Functional identities with pair of generalized semiderivations

(57-74)

4.1 Introduction

4.2 Preliminary results

4.3 Extension of Posner’s Theorems

4.4 Product of generalized semiderivations and commutativity of prime near rings

Chapter 5 Functional identities with traces of generalized n-derivations

(75-102)

5.1 Introduction


5.3 Traces of generalized 3-derivations and commutativity of prime near rings

Bibliography (103-120)

Acknowledgement

All praises and thanks to the Almighty, the most beneficent, the most

merciful, Who bestowed upon me the courage, patience and strength to embark

upon this work and carry it to its completion.

It has been my profound previlege to have accomplished my Ph.D. Thesis

under the able guidance of Dr. Asma Ali, Professor, Department of Mathemat-

ics, Aligarh Muslim University, Aligarh. I have immense pleasure in taking this

opportunity of acknowledge my deep sense of gratitude and highly indebtedness

to her, to whom I owe more than I can possibly express for her inspiring super-

vision, constant help, invaluable suggestions and encouragement to complete the

work. She has been inspiring source with unquestionable intellect and academic

excellence. The critical comments, she rendered during the work have gone a long

way in my understanding and presentation of the contents of this thesis. I wish,

I could be as perfect and determined like her in my future endeavour.

I extend my sincere thanks to Prof. M. Mursaleen, Chairman, Depart-

ment of Mathematics, Aligarh Muslim University, Aligarh, for providing me all

the departmental facilities whenever needed.

I have no words to express my gratitude and thanks to my parents

Mr. Karamat Ali and Mrs. Nasreen Begum for their limitless sacrifices to enrich

my future. They were always with me in good as well as in bad times in order

to keep me focussed towards my goal. I would like to express my special thanks

i

Preface

The present thesis entitled “Study of Functional identities with

derivations in near rings” includes a part of research work carried out by

the author under the able guidance of Prof. Asma Ali, at the Department of

Mathematics, Aligarh Muslim University, Aligarh. The thesis comprises

five chapters and each chapter is subdivided into various sections. The definitions,

examples, results and remarks etc. have been specified with double decimal num-

bers. The first figure denotes the chapter, the second represents the section in

the chapter and third points out the number of the definition, the example, the

result or the remark as the case may be in the particular chapter. For example

Theorem 4.2.3 refers to the third theorem appearing in the second section of the

fourth chapter.

Chapter 1 of the thesis contains some preliminary notions, basic definitions

and well known results which may be needed for the development of the subse-

quent text. This chapter as a matter of fact, aims at making the present thesis as

self contained as possible. However, the basic knowledge of the near ring theory

has been presumed and no attempt is made to include the proof of the results in

this chapter.

In 1987 Howard E. Bell and Mason [51] initiated the study of derivations

in near rings. A mapping d : N −→ N is said to be a derivation on a near

ring N if (i) d(x + y) = d(x) + d(y) and (ii) d(xy) = xd(y) + d(x)y holds for

all x, y ∈ N . It was shown by Wang [189] that condition (ii) is equivalent to

iii

d(xy) = d(x)y + xd(y) for all x, y ∈ N , which facilitates the study of derivations

in near rings.

In 1991 Bresar [69] introduced the notion of generalized derivation in rings.

As a motivation Golbasi [99] defined a generalized derivation in near rings. An

additive mapping F : N −→ N is said to be a right generalized (resp. left gen-

eralized) derivation with associated derivation d if F (xy) = F (x)y+ xd(y) (resp.

F (xy) = d(x)y + xF (y)) for all x, y ∈ N and F is said to be a general-

ized derivation with associated derivation d on N if it is both a right gen-

eralized derivation and a left generalized derivation on N with associated

derivation d. All derivations are generalized derivations.

In 1983, Bergen [54] introduced the notion of semiderivation in rings. Very

recently Asma et.al. [15] defined semiderivations in near rings. An additive map-

ping f : N −→ N is said to be a semiderivation on a near ring N if there exists a

function g : N −→ N such that (i) f(xy) = f(x)g(y) +xf(y) = f(x)y+ g(x)f(y)

and (ii) f(g(x)) = g(f(x)), for all x, y ∈ N . In case g is the identity map on

N , f is of course just a derivation on N , so notion of semiderivation generalizes

that of derivation. We define generalized semiderivation in near rings as follows:

An additive mapping F : N → N is said to be a generalized semiderivation on a

near ring N if there exists a semiderivation d : N −→ N associated with a map

g : N −→ N such that (i)F (xy) = F (x)y + g(x)d(y) = d(x)g(y) + xF (y) and

(ii)F (g(x)) = g(F (x)) for all x, y ∈ N .

In chapter 2 we study functional identities involving generalized semideriva-

tions in near rings. Section 2.2 starts with some preliminary results which are

necessary for developing the proof of the main theorems. In section 2.3 we

iv

extend some results of Bell and Martindale [50] for generalized semiderivations of

a prime near ring. Finally we investigate commutativity of the prime near ring N

admitting a generalized semiderivation F satisfying [F (U), F (U)] = {0}, where

U is a semigroup ideal of N .

In chapter 3 we discuss derivations which act as homomorphism or antiho-

momorphism, a study initiated by Bell and Kappe [46]. In section 3.2 a study of

generalized semiderivation has been made which acts as a homomorphism or an

antihomomorphism on a semigroup ideal of a prime near ring. Finally we obtain

commutativity of the prime near ring N with a generalized semiderivation F sat-

isfying one of the following conditions (i) F ([u, v]) = [u, v]; (ii) F ([u, v]) = −[u, v]

(iii) F (u ◦ v) = 0; (iv) F ([u, v]) = [F (u), v]; (v) F ([u, v]) = −[F (u), v];

(vi) F ([u, v]) = [u, F (v)] and (vii) F ([u, v]) = −[u, F (v)] for all u, v ∈ U ;

a nonzero semigroup ideal of N .

Chapter 4 deals with the study of functional identities involving pair of

generalized semiderivations. In section 4.2 we prove some preliminary results

which are required to prove our main theorems. In section 4.3 we extend

the two well-known Theorems of Posner [166, Theorem 1 and Theorem 2] in

case of generalized semiderivations of a prime near ring. Finally we prove

that a prime near ring N with generalized semiderivations F1 and F2 satisfy-

ing F1(x)F2(y)+F2(y)F1(x) ∈ Z for all x, y ∈ U , where U is a nonzero semigroup

ideal of N is a commutative ring.

In chapter 5 we study permuting generalized n-derivations in a near rings.

We concentrate on the functional identities involving traces of generalized

n-derivations. Ozturk and Yazarli [161, Theorem 3] proved that if N is a 2-torsion

v

free 3-prime near ring and D1, D2 are nonzero symmetric bi-derivations of N with

traces d1 and d2 respectively such that d2(y), d2(y)+d2(y) ∈ C(D1(x, z)), the cen-

tralizer of D1(x, z) for all x, y, z ∈ N , then (N,+) is abelian and d2(N) ⊆ Z,

the center of N . Further in [160, Theorem 5] they considered the permut-

ing generalized 3-derivation F of N with associated 3-derivation D satisfying

f(x), f(x) + f(x) ∈ C(D(y, z, w)) for all w, x, y, z ∈ N , where f is the trace of

F and concluded that (N,+) is abelian and f(N) ⊆ Z. We extend the above

results for a semigroup ideal of a prime near ring N .

Finally we prove the following theorem: Let N be a 3!-torsion free 3-prime

near ring and U be a nonzero additive subgroup and a semigroup ideal of N .

Suppose ∆ is a 3-derivation on N and F is a nonzero permuting generalized 3-

derivation of N associated with ∆ such that f(U) ⊆ U and δ(U) ⊆ U , where

f and δ are the trace of F and trace of ∆ respectively. If f(x), f(x) + f(x) ∈

C(F (u, v, w)), for all u, v, w, x ∈ U , then N is a commutative ring.

In the end, an exhaustive bibliography of the existing material related to the

subject matter of the thesis is included which may serve as source material for

those interested in the domain of this research area.

vi

Paper(s) Published/Accepted/Communicated for

Publication

[1] Abdelkarim Boua, A. Raji, Asma Ali and Farhat Ali, On generalized semideriva-

tions of prime near rings, Internat. J. Math. Math. Sci. 2015 Article ID 867923,

7 pages.

[2] Asma Ali, Mehsin Jabel Atteya, Phool Miyan and Farhat Ali, Semigroup ideals

and permuting 3-generalized derivations in prime near rings, Italian journal of

pure and applied Mathematics 35 (2015), 207-226.

[3] Asma Ali, Clauss Haetinger, Phool Miyan and Farhat Ali, Semigroup ideals and

permuting 3-derivations in prime near rings, Research Volume, Semigroups, Al-

gebras and Operator Theory, (2014), 67-79, Springer Press.

[4] Asma Ali, Basudeb Dhara, Shahoor Khan and Farhat Ali, Multiplicative

(generalized)-derivations and left ideals in semiprime rings, Hacettepe J. Math.

Stats., 44(6), (2015), 1293-1306.

[5] Asma Ali and Farhat Ali, Products of generalized semiderivations in prime near

rings, Research Volume, Advances in Algebra and Applications, Springer Press

(To appear).

[6] Asma Ali, Abdelkarim Boua and Farhat Ali, Semigroup ideals and generalized

semiderivations of prime near rings, Communicated to Afrika Mat..

[7] Asma Ali, Farhat Ali, Inzamam Ul Huque and Ambreen Bano On generalized

n-semiderivations of Prime near rings, Communicated to Bull. Austral. Math.

Soc..

Chapter 1

Preliminaries

1.1 Introduction

This chapter contains basic definitions and fundamental results in near rings

theory which we shall need for the development of the subject in the subse-

quent chapters of the present thesis. Of course, the knowledge of the elementary

algebraic concepts as those of near rings, subnear rings, near fields and homo-

morphisms etc. has been presumed. The material for the present chapter has

been collected mostly from the standard books like Clay[85], Meldrum[150] and

Pilz[165].

1.2 Some near ring theoretic concepts

This section is aimed to collect some important terminology in near ring theory.

Definition 1.2.1 (Near ring) A left near ring N is a triple (N,+, ?) with two

binary operations + and ? such that

(i) (N,+) is a group (not necessarily abelian).

(ii) (N, ?) is a semigroup.

(iii) a ? (b+ c) = a ? b+ a ? c, for all a, b, c ∈ N .

Analogously, if instead of (iii), we have the right distributive law

(iii)′ (a+ b) ? c = a ? c+ b ? c, for all a, b, c ∈ N

holds, then N is said to be a right near ring.

1

As in both the cases, the theory of near rings runs completely parallel, we

may consider left near rings throughout and for simplicity call them as near rings.

Example 1.2.1 (i) The most natural example of a left near ring is the set of all

identity preserving mappings acting from left of an additive group G (not neces-

sarily abelian) into itself with pointwise addition and composition of mappings as

multiplication. If these mappings act from right on G, then we get a right near

ring.

(ii) N = {0, a} with addition + and multiplication ? table defined as follows:

+ 0 a0 0 aa a 0

? 0 a0 0 aa 0 a

It is easily checked that (N,+, ?) is a left near ring.

(iii) Let (N,+) be any group. Define multiplication on N by a · b = b for all

a, b ∈ N . Then (N,+, ·) is a left near ring and it is known as a constant near

ring.

(iv) Let R be a ring and let R[x] be the set of all polynomials in one indetermi-

nate over R. Define addition in R[x] in the usual way and define composition by

f ◦ g = fg where f, g ∈ R[x]. Then (R[x],+, .) is a left near ring as well as a

right near ring.

(v) For more examples one may consult [84].

2

Definition 1.2.2 (Subnear ring) A nonvoid subset S of a near ring (N,+, ?)

is said to be a subnear ring of N if (S,+) is a subgroup of (N,+) and (S, ?) is a

subsemigroup of (N, ?).

Example 1.2.2 Let S be the set of all polynomials of the form a + bx from

R[x]. Then S is a subnear ring of R[x]. We identify a + bx by (a, b) ∈ R × R.

Then + and 0 of S induces the operations (a, b) + (c, d) = (a + c, b + d) and

(a, b).(c, d) = (a+ bc, bd) on R×R so (R×R,+, .) is a subnear ring of R[x].

Remark 1.2.1 A nonvoid set S of a near ring N is a subnear ring of N if and

only if for s1, s2 ∈ S, s1 − s2 ∈ S and s1s2 ∈ S.

Definition 1.2.3 (Characteristic of a near ring) The least positive integer

n (if exists) such that nx = 0 for all x ∈ N is called the characteristic of the near

ring N which is generally expressed as charN = n. If no such positive integer

exists, then N is said to have characteristic zero.

Definition 1.2.4 (Torsion free element) An element x in a near ring N is

said to be n-torsion free if nx = 0, implies that x = 0. If nx = 0 implies that

x = 0, for every x ∈ N , then we say that N is n-torsion free.

Definition 1.2.5 (Nilpotent element) An element x of a near ring N is said

to be nilpotent if there exists a positive integer n such that xn = 0.

Definition 1.2.6 (Distributive element) An element x of a near ring N is

called distributive if (y + z)x = yx+ zx, for all elements x, y ∈ N .

3

Remark 1.2.2 In a near ring, 0 and the identity element 1 are distributive ele-

ments. Note that this definition describes a relationship between an element and

the near ring which contains it. An element may be distributive in a subnear ring

which contains it without being distributive in the whole near ring.

Example 1.2.3 Let G be a group. Then End G={α | α is an endomorphism of

G}. So End G ⊆Mo(G) near ring of all homomorphism of G and End G consists

of entirely of distributive elements.

Definition 1.2.7 (Distributive near ring) A near ring N is said to be dis-

tributive if all of its elements are distributive.

Remark 1.2.3 In any near ring N

(i) x0 = 0, for all x ∈ N , but not necessarily 0x = 0. However, if N is distribu-

tive, then 0x = 0.

(ii) x(−y) = −xy, for all x, y ∈ N , but not necessarily (−x)y = −xy. However,

if N is distributive, then (−x)y = −xy.

Example 1.2.4 Let (G,+) be a non-abelian group and (R,+, ·) be a ring. Let

N = G ⊕ R. Then (N,+) is a non-abelian group. Define multiplication ? in N

as follows:

(g, r) ? (g′, r′) = (0, rr′).

Then (N,+, ?) is a distributive near ring.

4

Example 1.2.5 LetN = {0, a, b, c, x, y} with addition and multiplication defined

as follows:

+ 0 a b c x y0 0 a b c x ya a 0 y x c bb b x 0 y a cc c y x 0 b ax x b c a y 0y y c a b 0 x

? 0 a b c x y0 0 0 0 0 0 0a 0 a a a 0 0b 0 a a a 0 0c 0 a a a 0 0x 0 0 0 0 0 0y 0 0 0 0 0 0

Then (N,+, ?) is a distributive near ring.

Definition 1.2.8 (Distributively generated near ring) A near ring N is said

to be a distributively generated near ring (d − g) if it contains a multiplicative

subsemigroup of distributive elements which generates the additive group (N,+)

of N .

Example 1.2.6 The near ring generated additively by all endomorphisms of a

group (G,+) (not necessarily abelian), is a distributively generated near ring.

Remark 1.2.4 Each element of a distributively generated near ring can be ex-

pressed as a finite sum of distributive elements.

Remark 1.2.5 Every distributive near ring is a distributively generated near

ring but not conversely.

Example 1.2.7 Let N = {0, a, b, c} with addition and multiplication tables

defined as below:

5

+ 0 a b c0 0 a b ca a 0 c bb b c 0 ac c b 0 a

? 0 a b c0 0 0 0 0a 0 a 0 ab 0 0 0 0c 0 c 0 c

It is easy to check that N is a near ring but N is not a distributive near ring.

Consider the set S = {0, a, c}. Then it is easy to check that (S, ?) is a subsemi-

group of (N, ?) of distributive elements which generates (N,+) with respect to

+ only and N is a distributively generated near ring.

Definition 1.2.9 (Division near ring or Near field) A division near ring

or a near field is a near ring in which the nonzero elements form a group under

multiplication.

Remark 1.2.6 Every division ring is a near field but converse need not be true

in general.

Example 1.2.8 The near ring in Example 1.2.1 (ii) is a near field but not a

division ring.

Definition 1.2.10 (Zero-symmetric near ring) A near ring N is called zero-

symmetric if 0x = 0, for all x ∈ N (Recall that left-distributivity yields x0 = 0).

Example 1.2.9 The near ring in Example 1.2.1 (i) is a zero-symmetric near

ring.

Example 1.2.10 Let N = {0, 1, 2, 3} with addition and multiplication tables

6

defined as below:

+ 0 1 2 30 0 1 2 31 1 2 3 02 2 3 0 13 3 0 1 2

? 0 1 2 30 0 0 0 01 0 3 2 12 0 0 0 03 0 1 2 3

It can be verified that (N,+, ?) is a zero-symmetric near ring.

Remark 1.2.7 A (d− g) near ring is always zero-symmetric.

Definition 1.2.11 (Zero-commutative near ring) A near ring N is called

zero-commutative if xy = 0 implies yx = 0, for all x, y ∈ N .


defined as below:

+ 0 a b c0 0 a b ca a 0 c bb b c 0 ac c b a 0

? 0 a b c0 0 0 0 0a 0 a 0 ab 0 0 0 0c 0 c 0 0

Then (N,+, ?) is a zero-commutative near ring.

Definition 1.2.12 (Additive commutator and Multiplicative commuta-

tor) For any pair of elements x, y in a near ring N , the additive commutator

7

denoted by (x, y) = x + y − x − y and multiplicative commutator denoted by

[x, y] = xy − yx.

Definition 1.2.13 (Multiplicative centre) Multiplicative centre of a near ring

N is the set of all those elements of N which commute with every element of N

under multiplication and is denoted by Z.

Definition 1.2.14 (Additive centre) The additive centre of a near ring N is

the set of all those elements of N which commute with every element of N under

addition is denoted by ξ(N).

Definition 1.2.15 (Centralizer) Let N be a near ring. For all x ∈ N ,

C(x) = {a ∈ N | ax = xa} denotes the centralizer of x in N .

Definition 1.2.16 (Near ring homomorphism) Let (N1,⊕, ?) and (N2,+, ·)

be two near rings. Then a mapping f : N1 −→ N2 is called a near ring homo-

morphism if

(i) f(r1 ⊕ r2) = f(r1) + f(r2)

(ii) f(r1 ? r2) = f(r1) · f(r2)

for all r1, r2 ∈ N1.

Remark 1.2.8 Image of a near ring under a near ring homomorphism is again a

near ring.

Definition 1.2.17 (Near ring antihomomorphism) Let (N1,⊕, ?) and

(N2,+, ·) be two near rings. Then a mapping f : N1 −→ N2 is called a near

ring antihomomorphism if

8

(i) f(s1 ⊕ s2) = f(s1) + f(s2)

(ii) f(s1 ? s2) = f(s2) · f(s1)

for all s1, s2 ∈ N1.

Definition 1.2.18 (Ideal) An ideal of a near ring N is defined to be a normal

subgroup I of (N,+) such that

(i) NI ⊆ I.

(ii) (x+ i)y − xy ∈ I, for all x, y ∈ N and i ∈ I.

Normal subgroups of (N,+) satisfying (i) are called the left ideals and satisfying

(ii) are called right ideals.

In case of a (d− g) near ring, the condition (ii) above may be replaced by

(ii)′ IN ⊆ I.

Remark 1.2.9 Ideals may also be defined as the kernels of a near ring homo-

morphism.

Example 1.2.12 Consider N =

{(a b0 c

)| a, b, c ∈ F

}, the near ring of 2× 2

upper triangular matrices over a near field F . Then A =

{(a 00 0

)| a ∈ F

},

B =

{(0 b0 c

)| b, c ∈ F

}are left ideals of N and C =

{(a b0 0

)| a ∈ F

}is

a right ideal of N .


defined as below:

9


? 0 a b c0 0 0 0 0a 0 a 0 ab 0 0 b bc 0 a 0 c

It is easy to verify that A = {0, a} and B = {0, b} are ideals of N .

Definition 1.2.19 (Semigroup Ideal) A nonempty subset U of a near ring N

is said to be right (resp. left) semigroup ideal of N if UN ⊆ U (resp. NU ⊆ U)

and U is said to be a semigroup ideal if it is both a right semigroup ideal and a

left semigroup ideal of N .


defined as below:


? 0 a b c0 0 0 0 0a 0 0 a ab 0 a b bc 0 a c c

If we take A = {0, a}, B = {0, a, b} and C = {0, a, c}, then B,C are semigroup

right ideals of N and A is a semigroup ideal of N .

Definition 1.2.20 (Nilpotent Ideal) A right (left, two sided) ideal I of a near

ring N is said to be nilpotent if there exists a positive integer n > 1 such that

In = {0}.

10

Definition 1.2.21 (Prime Ideal) An ideal P of a near ring N is said to be

prime if for any ideals A,B in N , whenever AB ⊆ P , then either A ⊆ P or

B ⊆ P .

Example 1.2.15 In the near ring N of Example 1.2.1(iii) each normal subgroup

of (N,+) is a prime ideal of N .

Definition 1.2.22 (Completely Prime Ideal) An ideal P in a near ring N is

said to be completely prime if for any a, b ∈ N , ab ∈ P , implies that a ∈ P or

b ∈ P .

Definition 1.2.23 (Semiprime Ideal) An ideal P in a near ring N is said to

be semiprime if for any ideal A in N , A2 ⊆ P implies that A ⊆ P .

Remark 1.2.10 (i) A prime ideal is necessarily semiprime but converse need

not be true in general.

(ii) Intersection of prime (semiprime) ideals is semiprime.

Definition 1.2.24 (3-Prime near ring) A near ring N is said to be 3-prime if

zero ideal {0} is a prime ideal in N .

Remark 1.2.11 Equivalently a near ring N is 3-prime if and only if for any

a, b ∈ N , aNb = {0}, implies that either a = 0 or b = 0.

Example 1.2.16 Let C = {a+ ib | a, b ∈ R} be the set of all complex numbers.

Addition is the usual addition of complex numbers. Then (C,+) is a group. De-

11

fine multiplication ? on C by a ? b =| a | b. Then (C,+, ?) is a 3-prime near ring.

Example 1.2.17 The near ring in Example 1.2.1(iii) is a 3-prime near ring.

Definition 1.2.25 (3-Semiprime near ring) A near ring N is said to be 3-

semiprime if zero ideal {0} is a semiprime ideal in N .

Remark 1.2.12 Equivalently a near ring N is 3-semiprime if and only if for

a ∈ N , aNa = {0} implies that a = 0.

Remark 1.2.13 A near ring N is a 3-semiprime if and only if it has no nonzero

nilpotent ideals.

Example 1.2.18 Let N = {0, 1, 2, 3, 4, 5, 6, 7} be a near ring under addition

modulo 8 and multiplication defined as below:

? 0 1 2 3 4 5 6 70 0 0 0 0 0 0 0 01 0 1 0 1 1 1 1 12 0 2 0 2 2 2 2 23 0 3 0 3 3 3 3 34 0 4 0 4 4 4 4 45 0 5 0 5 5 5 5 56 0 6 0 6 6 6 6 67 0 7 0 7 7 7 7 7

Then (N,⊕8, ?) is a 3-semiprime near ring.

Motivated by the definition of a derivation in rings, Bell and Mason [51]

introduced the concept of a derivation in near rings.

12

Definition 1.2.26 (Derivation) A mapping d : N −→ N is said to be a deriva-

tion on a near ring N if for all x, y ∈ N

(i) d(x+ y) = d(x) + d(y) and

(ii) d(xy) = xd(y) + d(x)y

Example 1.2.19 Let V be a linear space with a basis {e1, e2, ...., en} over a field

F of characteristic different from two. Define multiplication · : V × V −→ V by

the rule v ·w = 0 for all v, w ∈ V with v 6∈ {e1,−e1} and e1w = w, (−e1)w = −w.

Then V is a zero-symmetric left near ring with respect to defined multiplication.

One can easily check that any linear transformation d : V −→ V such that

d(V ) ∩ {e1,−e1} = φ, is a derivation on V .

Example 1.2.20 Consider N = N1 ⊕ N2, where N1 is a zero symmetric near

ring and N2 is a commutative ring admitting a nonzero derivation δ. Define

d : N −→ N by d((u1, u2)) = (0, δ(u2)). Then d is a derivation on N .

It was shown by Wang [189] that in the definition 1.2.26 the condition (ii) is

equivalent to d(xy) = d(x)y + xd(y), for all x, y ∈ N , which facilitates the study

of derivations in near rings.

In 1991 Bresar [69] introduced the concept of a generalized derivation in

rings. Recently Golbasi [99] defined generalized derivation in near rings.

Definition 1.2.27 (Generalized derivation) Let d : N −→ N be a deriva-

tion on a near ring N . An additive mapping F : N −→ N is said to be a

right generalized (resp. left generalized) derivation with associated derivation d

13

if F (xy) = F (x)y+xd(y) (resp. F (xy) = d(x)y+xF (y)) for all x, y ∈ N and F is

said to be a generalized derivation with associated derivation d on N if it is both a

right generalized derivation and a left generalized derivation on N with associated

derivation d.

All derivations are generalized derivations.

Example 1.2.21 Let S be a zero-symmetric left near ring and let

N =

{(0 a0 b

)| a, b ∈ S

}.

Then N is a zero-symmetric left near ring under matrix addition and matrix

multiplication. Define maps d, F : N −→ N by

d

(0 a0 b

)=

(0 a0 0

)and

F

(0 a0 b

)=

(0 00 b

).

It can be verified that F is a right generalized derivation on N with associated

derivation d.

Example 1.2.22 Let S be a zero-symmetric left near ring and

N =

{(a b0 0

)| a, b ∈ S

}.

Define maps d, F : N −→ N by

14

d

(a b0 0

)=

(0 b0 0

)and

F

(a b0 0

)=

(a 00 0

).

Then F is a left generalized derivation on the zero symmetric left near ring N

with associated derivation d.

Example 1.2.23 Let S be a zero-symmetric left near ring and let

N =

{(a 0b c

)| a, b, c ∈ S

}.

Then N is a zero-symmetric left near ring. Define maps d, F : N −→ N by

d

(a 0b c

)=

(0 0b 0

)and

F

(a 0b c

)=

(0 0b 0

).

It can be checked that F is a right generalized derivation as well as a left

generalized derivation on N with associated derivation d. Hence F is a general-

ized derivation on N with associated derivation d.

Remark 1.2.14 Every derivation is a generalized derivation but not conversely.

Example 1.2.24 Let N =

{(a b0 c

)| a, b, c ∈ Z2

}. Define maps

F, d : N −→ N by

15

F

(a b0 c

)=

(a 00 0

)and

d

(a b0 c

)=

(0 b0 0

).

Then it can be verified that F is a generalized derivation on N but not a deriva-

tion on N .

1.3 Some key results

In this section we state some well-known results which may be frequently

referred in the subsequent text. For their proofs, the references are mentioned

against respective results for those who develop interest in them.

Theorem 1.3.1 (Neumann [157]) The additive group of a division near ring

is abelian.

Theorem 1.3.2 (Frohlich [95]) A (d − g) near ring N is distributive if and

only if N2 is additively commutative.

Theorem 1.3.3 (Frohlich [95]) A (d − g) near ring N with unity 1 is a ring

if (N,+) is abelian or if N is distributive.

Theorem 1.3.4 (Bell [37]) If a near ring N is zero-commutative, then the

following hold:

(i) ab = 0, implies axb = 0 for all x ∈ N .

(ii) The annihilator of any nonempty subset of N is an ideal.

(iii) The set N of all nilpotent elements is an ideal if and only if it is a subgroup

of (N,+).

16

Theorem 1.3.5 (Meldrum [150]) Let N be not necessarily zero symmetric

near ring. Let S be a semigroup of distributive elements of R. Then the distribu-

tively generated near ring generated by S is zero symmetric.

Theorem 1.3.6 (Meldrum [150]) If the near ring N is distributive, then the

elements of (N2,+) commute with each other.

Theorem 1.3.7 (Meldrum [150]) Let N be a zero symmetric near ring with

idempotents e and f such that ef = 0. Then r ∈ N has a decomposition

r = {(r − er)− f(r − er)}+ f(r − er) + er

Theorem 1.3.8 (Meldrum [150]) Let θ be a homomorphism from the near

ring N to the near ring S. Then

(i) θ is a group homomorphism from (N,+) to (S,+).

(ii) θ is a semigroup homomorphism from (N, .) to (S, .).

(iii) θ is a subnear ring of S.

Theorem 1.3.9 (Meldrum [150]) Let N be a near ring and K be an ideal of

R. If T is a subnear ring of N , then T +K is a subnear ring of N and

(T +K)/K = T/(T ∩K).

Theorem 1.3.10 (Bell [36]) Let N be a zero-commutative near ring having

no nonzero nilpotent elements. Then N contains a family of completely prime

ideals with trivial intersection.

17

Theorem 1.3.11 (Bell [36]) Let N be a zero-symmetric near ring having no

nonzero nilpotent elements. Then

(i) every distributive idempotent is central.

(ii) for every idempotent e and every element x ∈ N , ex2 = (ex)2.

(iii) if N has a multiplicative identity element, then all idempotents are central.

18

Chapter 2

Functional identities with general-ized semiderivations

2.1 Introduction

This chapter is devoted to the study of identities involving generalized

semiderivations in near rings. In 1983, Bergen [54] introduced the notion of

semiderivation in rings. Very recently Asma et.al. [15] defined semiderivations in

near rings. An additive mapping f : N −→ N is said to be a semiderivation on a

near ring N if there exists a function g : N −→ N such that (i) f(xy) = f(x)g(y)

+ xf(y) = f(x)y + g(x)f(y) and (ii) f(g(x)) = g(f(x)), for all x, y ∈ N . An

additive mapping F : N → N is said to be a generalized semiderivation on a

near ring N if there exists a semiderivation d : N −→ N associated with a map

g : N −→ N such that (i)F (xy) = F (x)y + g(x)d(y) = d(x)g(y) + xF (y) and

(ii)F (g(x)) = g(F (x)) for all x, y ∈ N .

Section 2.2 contains preliminary results which are necessary to develop the

proofs of various theorems in the chapter.

In section 2.3 we extend results of Bell and Martindale [50] for generalized

semiderivations of prime near ring.

Finally we establish some commutativity theorems in case of a prime near

ring N admitting a generalized semiderivation F satisfying [F (U), F (U)] = {0},

19

where U is a semigroup ideal of N . In fact the theorems that we prove extend

results of Bell and Mason [51, Theorem 2 and Theorem 3] and Bell [43, Theorem

2.1 and Theorem 4.1].


Motivated by a definition given by Bergen [54] for rings, Asma et.al. [15]

defined semiderivation in near rings.

Definition 2.2.1 (Semiderivation) An additive mapping d : N −→ N

is said to be a semiderivation on a near ring N if there exists a mapping

g : N −→ N such that (i) d(xy) = d(x)g(y) + xd(y) = d(x)y + g(x)d(y) and

(ii)d(g(x)) = g(d(x)) for all x, y ∈ N .

In case g is the identity map on N , d is of course just a derivation on N ,

so the notion of semiderivation generalizes that of derivation. Moreover, the

generalization is not trivial, as the following example shows :

Example 2.2.1 Let N = N1 ⊕N2, where N1 is a zero-symmetric near ring and

N2 is a ring. Then the map d : N −→ N defined by d((x, y)) = (0, y) is a

semiderivation associated with map g : N −→ N such that g(x, y) = (x, 0).

However d is not a derivation on N .

We define generalized semiderivation in near rings as follows:

Definition 2.2.2 (Generalized Semiderivation) An additive map-

ping F : N → N is said to be a generalized semiderivation on a near ringN if there

20

exists a semiderivation d : N −→ N associated with a map g : N −→ N such

that (i)F (xy) = F (x)y+g(x)d(y) = d(x)g(y)+xF (y) and (ii)F (g(x)) = g(F (x))

for all x, y ∈ N .

All semiderivations are generalized semiderivations. Moreover, if g is the

identity map on N , then all generalized semiderivations are merely generalized

derivations, again the notion of generalized semiderivation generalizes that of

generalized derivation and the generalization is not trivial.

Example 2.2.2 Let S be a 2-torsion free left near ring and let

N =

{ 0 x y0 0 00 0 z

| x, y, z ∈ S}.Define maps F, d, g : N → N by

F

0 x y0 0 00 0 z

=

0 xy 00 0 00 0 0

; d

0 x y0 0 00 0 z

=

0 0 y0 0 00 0 z

and

g

0 x y0 0 00 0 z

=

0 x 00 0 00 0 0

.It can be verified that N is a left near ring and F is a generalized semiderivation

with associated semiderivation d and a map g associated with d. However F is

not a generalized derivation on N .

Lemma 2.2.1 [41, Lemmas 1.3 and 1.4] Let N be 3-prime near ring and U be

a nonzero semigroup ideal of N .

(i) If x, y ∈ N and xUy = {0}, then x = 0 or y = 0.

(ii) If x ∈ N and xU = {0}, or Ux = {0}, then x = 0.

21

(iii) If x ∈ N centralizes U , then x ∈ Z.

Lemma 2.2.2 [41, Lemma 1.2] Let N be 3-prime near ring.

(i) If z ∈ Z\{0}, then z is not a zero divisor.

(ii) If Z\{0} contains an element z for which z + z ∈ Z, then (N,+) is abelian.

(iii) If z ∈ Z\{0} and x is an element of N such that xz ∈ Z, then x ∈ Z.

Lemma 2.2.3 [41, Lemma 1.5] If N is a 3-prime near ring and Z contains a

nonzero semigroup ideal, then N is a commutative ring.

We shall make use of the following well-known results, Propositions 2.2.4 -

2.2.9 may be found in [15]. However, in order to make our subsequent text as

much self contained as possible, we are giving the sketches of their proofs for left

near rings that are at times simpler and straight forward.

Proposition 2.2.4 Let N be a 3-prime near ring and U be a nonzero semigroup

ideal of N . If N admits a nonzero semiderivation d of N associated with a map

g such that g(U) = U , then d 6= 0 on U .

Proof. Let d(u) = 0, for all u ∈ U . Replacing u by xu, we get d(xu) = 0, for

all x ∈ N and u ∈ U . Thus d(x)g(u) + xd(u) = 0, for all x ∈ N and u ∈ U , i.e.,

d(x)g(u) = 0. The result follows by Lemma 2.2.1(ii).

Proposition 2.2.5 LetN be a 3-prime near ring admitting a nonzero semideriva-

tion d with a map g such that g(xy) = g(x)g(y) for all x, y ∈ N . Then N satisfies

22

the following partial distributive law:

(d(x)y + g(x)d(y))z = d(x)yz + g(x)d(y)z for all x, y, z ∈ N.

Proof. Let x, y, z ∈ N , by defining d we have

d(xyz) = d(xy)z + g(xy)d(z)

= (d(x)y + g(x)d(y))z + g(x)g(y)d(z). (2.2.1)

On the other hand,

d(xyz) = d(x)yz + g(x)d(yz)

= d(x)yz + g(x)(d(y)z + g(y)d(z))

= d(x)yz + g(x)d(y)z + g(x)g(y)d(z). (2.2.2)

Combining (2.2.1) and (2.2.2), we obtain

(d(x)y+g(x)d(y))z+g(x)g(y)d(z) = d(x)yz+g(x)d(y)z+g(x)g(y)d(z) for all x, y, z ∈ N.

(d(x)y + g(x)d(y))z = d(x)yz + g(x)d(y)z for all x, y, z ∈ N.


ideal of N . If N admits a nonzero semiderivation d of N associated with a map

g such that g(uv) = g(u)g(v) for all u, v ∈ U . If a ∈ N and ad(U) = 0, (or

d(U)a = 0), then a = 0.

Proof. Let ad(u) = 0, for all u ∈ U . Replacing u by uv, we get ad(uv) = 0, for all

u, v ∈ U or a(d(u)g(v)+ud(v)) = 0, for all u, v ∈ U , we get ad(u)g(v)+aud(v) =

0, for all u, v ∈ U or aud(v) = 0, for all u, v ∈ U . Choosing v such that d(v) 6= 0

23

and applying Lemma 2.2.1(i), we get a = 0.

Also, let d(v)a = 0, for all v ∈ U . Replacing v by uv, we get d(uv)a = 0, for all

u, v ∈ U or (d(u)v+ g(u)d(v))a = 0, for all u, v ∈ U . Using Proposition 2.2.5, we

get d(u)va + g(u)d(v)a = 0, for all u, v ∈ U or d(u)va = 0 or d(u)va = 0, for all

u, v ∈ U . Choosing u such that d(u) 6= 0 and applying Lemma 2.2.1(i), we get

a = 0.

Proposition 2.2.7 Let N be a 2-torsion free 3-prime near ring and U be a

nonzero semigroup ideal of N . If d is a nonzero semiderivation of N associated

with a map g such that g(U) = U and g(uv) = g(u)g(v), for all u, v ∈ U , then

d2(U) 6= {0}.

Proof. Suppose d2(U) = {0}. Then for u, v ∈ U exploit the definition of d in

different ways to obtain

0 = d2(uv) = d(d(uv)) = d(d(u)v + g(u)d(v)) for all u, v ∈ U,

= d2(u)v + g(d(u))d(v) + d(g(u))d(v) + g(u)d2(v),

= d(g(u))d(v) + d(g(u))d(v).

Note that g(d(u)) = d(g(u)) and g(U) = U , we get

2d(u)d(v) = 0 for all u, v ∈ U.

Since N is a 2-torsion free, we get

d(u)d(v) = 0 for all u, v ∈ U. (2.2.3)

Replacing v by wv in (2.2.3), we get

d(u)d(wv) = 0 for all u, v, w ∈ U.

24

d(u)(d(w)v + g(w)d(v)) = 0 for all u, v, w ∈ U.

d(u)d(w)v + d(u)g(w)d(v) = 0 for all u, v, w ∈ U.

This implies that

d(u)g(w)d(v) = 0 for all u, v, w ∈ U.

d(u)wd(v) = 0 for all u, v, w ∈ U.

d(U)Ud(U) = {0}.

Thus we obtain that d = 0, a contradiction.


ideal of N . Suppose d is a nonzero semiderivation of N associated with a map g

such that g(uv) = g(u)g(v), for all u, v ∈ U . If d(U) ⊆ Z, then N is a commuta-

tive ring.

Proof. We begin by showing that (N,+) is abelian, which by Lemma 2.2.2(ii) is

accomplished by producing z ∈ Z\{0} such that z + z ∈ Z. Let a be an element

of U such that d(a) 6= 0. Then for all x ∈ N , ax ∈ U and ax + ax = a(x + x) ∈

U , so that d(ax) ∈ Z and d(ax) + d(ax) ∈ Z; hence we need only show that

there exists x ∈ N such that d(ax) 6= 0. Suppose this is not the case, so that

d((ax)a) = 0 = d(ax)g(a) +axd(a) = axd(a) for all x ∈ N . Since d(a) is not zero

divisor by Lemma 2.2.2(i), we get aN = {0}, so that a = 0 - a contradiction.

Therefore (N,+) is abelian as required.

We are given that [d(u), x] = 0 for all u ∈ U and x ∈ N . Replacing u by uv,

we get [d(uv), x] = 0, which yields [d(u)v + g(u)d(v), x] = 0 for all u, v ∈ U and

x ∈ N . Since (N,+) is abelian and d(U) ⊆ Z, we have

d(u)[v, x] + d(v)[x, g(u)] = 0 for all u, v ∈ U and x ∈ N. (2.2.4)

25

Replacing x by g(u), we obtain d(u)[v, g(u)] = 0 for all u, v ∈ U ; and choosing

u ∈ U such that d(u) 6= 0 and applying Lemma 2.2.1(iii), we get g(u) ∈ Z. It

then follows from (2.2.4) that d(u)[v, x] = 0 for all v ∈ U and x ∈ N ; therefore

U ⊆ Z and Lemma 2.2.3 completes the proof.


nonzero semigroup ideal of N . Suppose that d is a nonzero semiderivation of

N associated with a map g such that g(U) = U and g(uv) = g(u)g(v) for all

u, v ∈ U . If [d(U), d(U)] = {0}, then N is a commutative ring.

Proof. We are given that [d(U), d(U)] = {0}. Then d(u)d(vd(w)) =

d(vd(w))d(u), for all u, v, w ∈ U , i.e., d(u)(d(v)g(d(w)) + vd2(w)) =

(d(v)g(d(w))+vd2(w))d(u), for all u, v, w ∈ U . Then by Proposition 2.2.5, we get

d(u)d(v)g(d(w))+d(u)vd2(w)) = d(v)g(d(w))d(u)+vd2(w)d(u);d(u)d(v)d(g(w))+

d(u)vd2(w)) = d(v)d(g(w))d(u) + vd2(w)d(u); d(u)d(v)d(w) + d(u)vd2(w)) =

d(v)d(w)d(u) + vd2(w)d(u) for all u, v, w ∈ U and since [d(U), d(U)] = {0},

we obtain

d(u)vd2(w)) = vd2(w)d(u) for all u, v, w ∈ U. (2.2.5)

Replace v by xv in (2.2.5), to get

d(u)xvd2(w)) = xvd2(w)d(u) for all u, v, w ∈ U and x ∈ N.

Using (2.2.5), the above relation yields that d(u)xvd2(w)) = xd(u)vd2(w), for all

u, v, w ∈ U and x ∈ N , i.e.,[d(u), x]vd2(w) = 0, for all u, v, w ∈ U and x ∈ N by

Proposition 2.2.5. Thus [d(u), x]Ud2(w) = 0, for all u,w ∈ U and x ∈ N . Since

d2(U) 6= 0 by Proposition 2.2.7, Lemma 2.2.1(i) gives d(U) ⊆ Z, and the result

follows by Proposition 2.2.8.

26

Proposition 2.2.10 Let N be a 3-prime near ring admitting a generalized

semiderivation F associated with a semiderivation d. If g is an onto map asso-

ciated with d such that g(xy) = g(x)g(y) for all x, y ∈ N , then N satisfies the

following partial distributive laws:

(i) (F (x)y + g(x)d(y))z = F (x)yz + g(x)d(y)z for all x, y, z ∈ N .

(ii) (d(x)g(y) + xF (y))z = d(x)g(y)z + xF (y)z for all x, y, z ∈ N .

Proof. (i) Let x, y, z ∈ N , by defining F we have

F (xyz) = F (xy)z + g(xy)d(z)

= (F (x)y + g(x)d(y))z + g(x)g(y)d(z). (2.2.6)

On the other hand,

F (xyz) = F (x)yz + g(x)d(yz)

= F (x)yz + g(x)(d(y)z + g(y)d(z))

= F (x)yz + g(x)d(y)z + g(x)g(y)d(z). (2.2.7)


(F (x)y+g(x)d(y))z+g(x)g(y)d(z) = F (x)yz+g(x)d(y)z+g(x)g(y)d(z) for all x, y, z ∈ N.

(F (x)y + g(x)d(y))z = F (x)yz + g(x)d(y)z for all x, y, z ∈ N.

(ii) Let x, y, z ∈ N , by defining F we have

F (xyz) = F (xy)z + g(xy)d(z)

= (d(x)g(y) + xF (y))z + g(x)g(y)d(z). (2.2.8)

27

On the other hand,

F (xyz) = d(x)g(yz) + xF (yz)

= d(x)g(y)g(z) + x(F (y)z + g(y)d(z))

= d(x)g(y)z + xF (y)z + g(x)g(y)d(z). (2.2.9)


(d(x)g(y) + xF (y))z = d(x)g(y)z + xF (y)z for all x, y, z ∈ N.

2.3 Extension of Bell and Martindale’s results

In [50, Lemma 1] Bell and Martindale proved that if R is a prime ring, I a

nonzero ideal of R and f : R −→ R is a nonzero semiderivation, then f is nonzero

on I. We extend the result for a generalized semiderivation of 3-prime near ring

in case of a semigroup ideal.

Theorem 2.3.1 Let N be a 3-prime near ring and U be a nonzero semigroup

ideal of N . If F is a nonzero generalized semiderivation of N with associated

semiderivation d and a map g associated with d such that g(U) = U , then F 6= 0

on U .

Proof. Let F (u) = 0 for all u ∈ U . Replacing u by ux, we get F (ux) = 0 for

all u ∈ U and x ∈ N . Thus

F (u)x+ g(u)d(x) = 0 = Ud(x) for all x ∈ N

and it follows by Lemma 2.2.1(ii) that d = 0. Therefore, we have

F (xu) = F (x)u = 0 for all u ∈ U for all x ∈ N

28

and another appeal to Lemma 2.2.1(ii) gives F = 0, which is a contradiction.

Following theorems extend Lemma 2 and Lemma 3 of the mentioned paper of

Bell and Martindale.


ideal of N . Suppose that F is a nonzero generalized semiderivation of N with

associated semiderivation d and a map g associated with d such that g(U) = U

and g(uv) = g(u)g(v) for all u, v ∈ U . If a ∈ N and aF (U) = 0 (or F (U)a = 0)

then a = 0.

Proof. Suppose that aF (U) = {0}. Then for u, v ∈ U

aF (uv) = a(F (u)v + g(u)d(v)) = aF (u)v + ag(u)d(v)

= aud(v) = 0 for all u, v ∈ U and a ∈ N.

Thus by Lemma 2.2.1(i), a = 0 or d(U) = {0}. If d(U) = {0}, then

ad(u)g(v) + auF (v) = 0 = auF (v) for all u, v ∈ U.

Since F (U) 6= {0}, it follows by Theorem 2.3.1 that a = 0.

Suppose that F (U)a = {0}. Then for u, v ∈ U

F (uv)a = (F (u)v + g(u)d(v))a = {0}.

Using Proposition 2.2.10(i), we get

F (u)va+ g(u)d(v)a = ud(v)a = 0 for all u, v ∈ U and a ∈ N.

29

Thus by Lemma 2.2.1 (i), a = 0 or d(U) = {0}. If d(U) = {0}, then

d(u)g(v)a+ uF (v)a = 0 = uF (v)a for all u, v ∈ U.

Since F (U) 6= {0}, it follows by Theorem 2.3.1 that a = 0.

Theorem 2.3.3 Let N be a 2-torsion free 3-prime near ring and U be a nonzero

semigroup ideal of N . If N admits a nonzero generalized semiderivation F

associated with a nonzero semiderivation d and a map g associated with d such

that g(U) = U and g(uv) = g(u)g(v) for all u, v ∈ U and F (U) ⊆ U , then F 2 6= 0

on U .

Proof. Let F 2(U) = {0}. Then for u, v ∈ U , we have

0 = F 2(uv) = F (F (uv)).

= F (F (u)v + g(u)d(v))

= F (F (u)v) + F (g(u)d(v))

= g(F (u))d(v) + F (u)d(v) + g(u)d2(v)

= F (g(u))d(v) + F (u)d(v) + ud2(v)

= F (u)d(v) + F (u)d(v) + ud2(v).

This implies that

2F (u)d(v) + ud2(v) = 0 for all u, v ∈ U. (2.3.1)

Replacing u by F (u) in (2.3.1), we get

2F (F (u))d(v) + F (u)d2(v) = 0 for all u, v ∈ U.

30

This implies that

2F 2(u)d(v) + F (u)d2(v) = 0.

F (u)d2(v) = 0.

By Theorem 2.3.2, we obtain that d2(v) = 0 or F (U) = {0}. If d2(v) = 0, then

d = 0 by Proposition 2.2.7, a contradiction. So, we find F (U) = {0}, again a

contradiction by Theorem 2.3.1.

2.4 Commutativity of prime near rings satisfyingcondition [F (U), F (U)] = {0}

Over the last two decades the literature on near rings contains a number of

results asserting that certain conditions implying commutativity in rings imply

multiplicative or additive commutativity in special classes of near rings. Many au-

thors established several commutativity theorems for near rings admitting a suit-

ably constrained derivation. In [51, Theorem 2] Bell and Mason proved that if N

is a 3-prime near ring admitting a nonzero derivation d such that d(N) ⊆ Z, then

N is commutative ring. Further in [43, Theorem 2.1], the first author obtained

the result replacing d by a generalized derivation. Motivated by the aforemen-

tioned results we prove the following theorems for a generalized semiderivation of

a near ring in the setting of a semigroup ideal.


ideal of N . Suppose that N admits a nonzero generalized semiderivation F with

associated semiderivation d and a map g associated with d such that g(U) = U

and g(uv) = g(u)g(v) for all u, v ∈ U . If F (U) ⊆ Z, then (N,+) is abelian.

Moreover, if N is 2-torsion free, then N is commutative ring.

31

Proof. We begin by showing that (N,+) is abelian, which by Lemma 2.2.2(ii) is

accomplished by producing z ∈ Z\{0} such that z + z ∈ Z. Let a be an element

of U such that F (a) 6= 0. Then for all x ∈ N , ax ∈ U and ax+ax = a(x+x) ∈ U ,

so that F (ax) ∈ Z and F (ax) + F (ax) ∈ Z; hence we need only to show that

there exists x ∈ N such that F (ax) 6= 0. Suppose that this is not the case, so

that F ((ax)a) = 0 = F (ax)a+ g(ax)d(a) = g(a)g(x)d(a) = axd(a) for all x ∈ N .

By Lemma 2.2.1(i) either a = 0 or d(a) = 0.

If d(a) = 0, then F (xa) = F (x)a + g(x)d(a); that is, F (xa) = F (x)a ∈ Z, for

all x ∈ N . Thus, [F (u)a, y] = 0 for all y ∈ N and u ∈ U . This implies that

F (u)[a, y] = 0 for all u ∈ U and y ∈ N and Lemma 2.2.2(i) gives a ∈ Z. Thus,

0 = F (ax) = F (xa) = F (x)a for all x ∈ N . Replacing x by u ∈ U , we have

F (U)a = 0, and by Lemma 2.2.2(i) and Theorem 2.3.1, we have a contradiction.

To complete the proof, we show that if N is 2-torsion free, then N is commuta-

tive.

Consider first case d = 0. This implies that F (ux) = F (u)x ∈ Z for all u ∈ U

and x ∈ N . By Theorem 2.3.1, we have u ∈ U such that F (u) ∈ Z\{0}, so N is

commutative by Lemma 2.2.2(iii).

Now consider the case d 6= 0. Let c ∈ Z\{0}. This implies that x ∈ U , F (xc) =

F (x)c+g(x)d(c) = F (x)c+xd(c) ∈ Z. Thus (F (x)c+xd(c))y = y(F (x)c+xd(c))

for all x, y ∈ U and c ∈ Z. Therefore, by Proposition 2.2.10(i), F (x)cy+xd(c)y =

yF (x)c + yxd(c) for all x, y ∈ U and c ∈ Z. Since d(c) ∈ Z and F (x) ∈ Z, we

obtain d(c)[x, y] = 0 for all x, y ∈ U and c ∈ Z. Let d(Z) 6= {0}. Choosing c such

that d(c) 6= 0 and noting that d(c) is not a zero divisor, we have [x, y] = 0 for all

32

x, y ∈ U . By Lemma 2.2.1(iii), U ⊆ Z; hence N is commutative by Lemma 2.2.3.

The remaining case is d 6= 0 and d(Z) = {0}. Suppose we can show that U ∩Z 6=

{0}. Taking z ∈ (U ∩Z)\{0} and x ∈ N , we have F (xz) = F (x)z ∈ Z; therefore

F (N) ⊆ Z by Lemma 2.2.2(iii). Let F (x) ∈ Z for all x ∈ N.

Since d(Z) = 0, for all x, y ∈ N . We have

0 = d(F (xy)).

0 = d(F (x)y + g(x)d(y)).

0 = F (x)d(y) + g(x)d2(y) + d(g(x))g(d(y)) for all x, y ∈ Z.

Hence F (xd(y)) = −d(g(x))g(d(y)) ∈ Z for all x, y ∈ N . By hypothesis, we have

d(x)d(y) ∈ Z for all x, y ∈ N . This implies that

d(x)(d(x)d(y)− d(y)d(x)) = 0 for all x, y ∈ N.

Left multiplying by d(y), we arrive at

d(y)d(x)N(d(x)d(y)− d(y)d(x)) = {0} for all x, y ∈ N.

Since N is a 3-prime near ring, we get

[d(x), d(y)] = 0 for all x, y ∈ N.

Using Proposition 2.2.9, N is a commutative ring.

Assume that U ∩ Z = {0}. For each u ∈ U , F (u2) = F (u)u + g(u)d(u) =

F (u)u+ ud(u) = u(F (u) + d(u)) ∈ U ∩ Z. So F (u2) = 0, thus for all u ∈ U and

x ∈ N , F (u2x) = F (u2)x + g(u2)d(x) = u2d(x) ∈ U ∩ Z. So u2d(x) = 0 and

Proposition 2.2.6, u2 = 0. Since F (xu) = F (x)u+g(x)d(u) = F (x)u+xd(u) ∈ Z

for all u ∈ U and x ∈ N . We have (F (x)u + xd(u))u = u(F (x)u + xd(u)) and

33

right multiplying by u gives uxd(u)u = 0. Consequently, d(u)uNd(u)u = {0}.

So that d(u)u = 0 for all u ∈ U , so F (u)u = 0 for all u ∈ U . But by Theorem

2.3.1, there exist u0 ∈ U for which F (u0) 6= 0; and F (u0) ∈ Z, we get u0 = 0,

contradiction. Therefore, U ∩ Z 6= {0} as required.

Herstein [111] proved that a prime ring R with a nontrivial derivation d sat-

isfying the condition [d(x), d(y)] = 0 for all x, y ∈ R must be commutative. Bell

and Mason [51, Theorem 3] investigated a parallel result in case of near rings.

Further Bell [43, Theorem 4.1] proved the above result for a generalized derivation

of a 3-prime near ring which states that if N is a 2-torsion free 3-prime near ring

and F : R −→ R is nonzero generalized derivation such that [F (N), F (N)] = {0},

then N is a commutative ring. Now, we prove the following theorems replacing

F by a generalized semiderivation which extend these results:


semigroup ideal of N . Suppose that N admits a nonzero generalized semideriva-

tion F associated with a semiderivation d and a map g associated with d such

that g(uv) = g(u)g(v) for all u, v ∈ U and g(U) = U . If [F (U), F (U)] = {0},

then (N,+) is abelian.

Proof. Assume that x ∈ U such that

[x, F (U)] = [x+ x, F (U)] = 0 and u, v ∈ U such that u+ v ∈ U.

Then

[x+ x, F (u+ v)] = 0.

(x+ x)F (u+ v) = F (u+ v)(x+ x).

(x+ x)(F (u) + F (v)) = F (u+ v)x+ F (u+ v)x.

34

(x+ x)F (u) + (x+ x)F (v) = xF (u+ v) + xF (u+ v).

F (u)(x+ x) + F (v)(x+ x) = x(F (u) + F (v)) + x(F (u) + F (v)).

F (u)(x+ x) + F (v)(x+ x) = xF (u) + xF (v) + xF (u) + xF (v).

xF (u) + xF (u) + F (v) + xF (v) = xF (u) + xF (v) + xF (u) + xF (v).

xF (u) + xF (v) = xF (v) + xF (u).

i.e.,

xF (u) + xF (v)− xF (v)− xF (u) = 0.

x(F (u) + F (v)− F (v)− F (u)) = 0.

xF (u+ v − u− v) = 0 for all u, v, x ∈ U.

That gives that xF (c) = 0, where c = u+ v − u− v. Let a, b ∈ U . Then ab ∈ U

and ab+ab = a(b+ b) ∈ U . Since [F (U), F (U)] = {0}, we take x = F (ab) so that

[F (ab) + F (ab), F (u+ v)] = {0}

which gives F (c)F (ab) = 0, i.e., F (c)F (U2) = 0

Since U2 is also a semigroup ideal. By Theorem 2.3.2, F (c) = 0, i.e.,

F (u+ v − u− v) = 0 for all u, v ∈ U such that u+ v ∈ U. (2.4.1)

Replacing u by ry and v by rz for y, z ∈ N and u, r ∈ U , we have u, v ∈ U and

u+ v = ry + rz = r(y + z) ∈ U .

Then (2.4.1) implies that

F (ry + rz − ry − rz) = 0 for all r ∈ U and y, z ∈ N. (2.4.2)

35

Now replace r by wr, to get

F (wry + wrz − wry − wrz) = 0 for all r, w ∈ U and y, z ∈ N.

F (w(ry + rz − ry − rz)) = 0 for all r, w ∈ U and y, z ∈ N.

This implies that

d(w)g(ry + rz − ry − rz) + wF (ry + rz − ry − rz) = 0.

Using (2.4.2), we obtain

d(w)g(ry + rz − ry − rz) = 0 for all r, w ∈ U and y, z ∈ N.

d(U)g(ry + rz − ry − rz) = {0} for all r ∈ U and y, z ∈ N.

Application of Proposition 2.2.6 gives that

(ry + rz − ry − rz) = 0 for all r ∈ U and y, z ∈ N.

r(y + z − y − z) = 0 for all r ∈ U and y, z ∈ N.

U(y + z − y − z) = {0} and y, z ∈ N.

Now using Lemma 2.2.1(ii), we get

y + z − y − z = 0 for all y, z ∈ N.

Hence (N,+) is abelian.


semigroup ideal of N . Suppose that N admits a nonzero generalized semideriva-

tion F with associated nonzero semiderivation d and a map g associated with d

36

such that g(uv) = g(u)g(v) for all u, v ∈ U and g(U) = U . If [F (U), F (U)] = {0},

then N is a commutative ring.

Proof. By hypothesis [F (U), F (U)] = {0}. Then

F (u)F (F (w)v) = F (F (w)v)F (u) for all u, v, w ∈ U.

F (u)(d(F (w))g(v) + F (w)F (v)) = (d(F (w))g(v) + F (w)F (v))F (u).

Using Proposition 2.2.10(ii), we get

F (u)d(F (w)g(v) + F (u)F (w)F (v) = d(F (w))g(v)F (u) + F (w)F (v)F (u).

Then we have

F (u)d(F (w)g(v) = d(F (w))g(v)F (u) for all u, v, w ∈ U. (2.4.3)

Thus

F (u)d(F (w)v = d(F (w))vF (u). (2.4.4)

Replacing v by vt for all t ∈ N and using (2.4.4), we get

d(F (w)vF (u)t = d(F (w))vtF (u), i.e.

d(F (w))v[F (u), t] = {0} for all u, v, w ∈ U and t ∈ N.

d(F (w))U [F (u), t] = {0} for all u,w ∈ U and t ∈ N.

By Lemma 2.2.1(i), we get either [F (u), t] = 0 or d(F (w)) = 0. In the first

case F (U) ⊆ Z and Theorem 2.4.1 completes the proof. Let us assume that

d(F (U)) = {0}. Then 0 = d(F (uv)) = d(F (u)v + g(u)d(v)) = d(F (u)v) +

d(ud(v)) = F (u)d(v) + d(u)g(d(v)) + ud2(v) = F (u)d(v) + d(u)d(v) + ud2(v) for

all u, v ∈ U . we have

F (u)d(v) + d(u)d(v) + ud2(v) = 0. (2.4.5)

37

Now replacing u by uw, we get

F (uw)d(v) + d(uw)d(v) + uwd2(v) = 0.

(d(u)g(w) + uF (w))d(v) + (d(u)g(w) + ud(w))d(v) + uwd2(v) = 0.

Using Proposition 2.2.5 and Proposition 2.2.10(ii), we have

d(u)g(w)d(v) + uF (w)d(v) + d(u)g(w)d(v) + ud(w)d(v) + uwd2(v) = 0.

2d(u)wd(v) + u{F (w)d(v) + d(w)d(v) + wd2(v)} = 0.


2d(u)wd(v) = 0.

This implies that

2d(u)Ud(v) = 0 for all u, v ∈ U.

Since N is 2-torsion free, we get

d(u)Ud(v) = {0}.

Thus, we obtain that d(U) = {0}, a contradiction, by Proposition 2.2.4.

38

Chapter 3

Functional identities with gener-alized semiderivations acting ashomomorphism and antihomo-morphism

3.1 Introduction

This chapter deals with the study of generalized semiderivations of near ring

N which act as homomorphism or antihomomorphism on a well behaved subset

of N . Let S be a nonempty subset of N and F be a generalized semiderivation

of N . If F (xy) = F (x)F (y) (resp. F (xy) = F (y)F (x)) for all x, y ∈ S, then

we say that F acts as a homomorphism (resp. as antihomomorphism) on S. In

[46] Bell and Kappe proved that if R is a semiprime ring and d is a derivation of

R which is either an endomorphism or an antiendomorphism on R, then d = 0.

Asma et.al. in [16] proved the result for (θ, φ)-derivation d of a prime ring R in

case of a Lie ideal of R.

In section 3.2 we investigate similar results for a prime near ring N with a

generalized semiderivation F of N which is a homomorphism or an antihomomor-

phism on a nonzero semigroup ideal of N and conclude that F is zero or identity

map or near ring a commutative ring.

In section 3.3 we obtain commutativity of prime near ring N admitting a

39

generalized semiderivation F satisfying any one of the following conditions:

(i) F ([u, v]) = [u, v]; (ii) F ([u, v]) = −[u, v]; (iii) F (u ◦ v) = 0;

(iv) F ([u, v]) = [F (u), v]; (v) F ([u, v]) = −[F (u), v]; (vi) F ([u, v]) = [u, F (v)];

(vii) F ([u, v]) = −[u, F (v)] for all u, v ∈ U ; a nonzero semigroup ideal of N .

3.2 Results involving generalized semideri-vations acting as homomorphism and antihomo-morphism

In 1989 Bell and Kappe [46] showed that if a derivation d of a prime ring R

acts as a homomorphism or an antihomomorphism on a nonzero right ideal of R,

then d = 0 on R. Gusic [104] proved more general result which states as follows:

Theorem 3.2.1 Let R be a prime ring, d any function on R (not necessary a

derivation nor an additive function). Let F be any function on R (not necessarily

additive) satisfying F (xy) = F (x)y + xd(y) for all x, y ∈ R, and I be a nonzero

ideal in R.

(a) If F acts as a homomorphism on I, a nonzero ideal of R, then d = 0 and

F = 0 or F is an identity map on N .

(b) If F acts as an antihomomorphism on I, a nonzero ideal of R, then d = 0

and F = 0 or F is an identity map on N and R is a commutative ring.

More recently Dhara [89] proved that if R is a semiprime ring and F is a

generalized derivation of R acting as a homomorphism or an antihomomorphism

on a two sided ideal I of R, then d(I) = 0 or R contains a nonzero central ideal.

40

Motivated by the above results we prove the similar results for a prime near

ring with a generalized semiderivation. More precisely we prove that a general-

ized semiderivation which acts as a homomorphism or an antihomomorphism on

a semigroup ideal of a 3-prime near ring is zero or identity map or near ring a

commutative ring.


ideal of N . Let F be a nonzero generalized semiderivation of N associated with

a semiderivation d and a map g associated with d such that g(uv) = g(u)g(v)

for all u, v ∈ U and g(U) = U . If F acts as a homomorphism on U , then F is

identity map on N and d = 0.

Proof. By the hypothesis,

F (xy) = d(x)g(y) + xF (y) = F (x)F (y) for all x, y ∈ U.

Replacing y by yz in the above relation, we obtain

F (x)F (yz) = d(x)g(yz) + xF (yz).

F (x)F (y)F (z) = d(x)g(yz) + x(d(y)g(z) + yF (z)).

F (xy)F (z) = d(x)g(yz) + x(d(y)g(z) + yF (z)).

(d(x)g(y) + xF (y))F (z) = d(x)g(yz) + x(d(y)g(z) + yF (z)).

Using Proposition 2.2.10(ii), we get

d(x)g(y)F (z) + xF (y)F (z) = d(x)g(yz) + x(d(y)g(z) + yF (z)) for all x, y, z ∈ U.

41

d(x)g(y)F (z) + xF (yz) = d(x)g(yz) + x(d(y)g(z) + yF (z)) for all x, y, z ∈ U.

d(x)g(y)F (z)+x(d(y)g(z)+yF (z)) = d(x)g(yz)+xd(y)g(z)+xyF (z)) for all x, y, z ∈ U.

d(x)g(y)F (z) + xd(y)g(z) + xyF (z) = d(x)g(yz) + xd(y)g(z) + xyF (z).

d(x)g(y)F (z) = d(x)g(y)g(z).

d(x)yF (z) = d(x)yz for all x, y, z ∈ U.

d(x)y(F (z)− z) = 0 for all x, y, z ∈ U.

d(x)U(F (z)− z) = {0} for all x, y, z ∈ U.

It follows by Lemma 2.2.1(i) either d(U) = {0} or F (z) = z for all z ∈ U . In

fact, as we now show both of these conditions hold. Suppose that F (u) = u for

all u ∈ U . Then for all u ∈ U and x ∈ N , F (xu) = xu = d(x)g(u) + xF (u) =

d(x)u+ xu, hence d(x)U = {0} for all x ∈ N and d = 0.

On the other hand, suppose that d(U) = {0}, so that d = 0. Then for all x, y ∈ U ,

F (xy) = F (x)y = F (x)F (y), so that F (x)(y − F (y)) = 0. Replacing y by zy,

z ∈ N and noting that F (zy) = zF (y), we see that F (x)N(y − F (y)) = {0} for

all x, y ∈ U . Therefore, F (U) = {0} or F is the identity on U . But F (U) = {0}

contradicts Theorem 2.3.1, so F is the identity on U .

Since F is the identity map on U and F (xy) = xF (y) for all x, y ∈ N ,

F (ux) = ux = uF (x) for all u ∈ U and x ∈ N . Thus U(x − F (x)) = {0}

for all x ∈ N . Hence F is the identity on N .


ideal of N . Let F be a nonzero generalized semiderivation of N associated with

42

a semiderivation d and a map g associated with d such that g(uv) = g(u)g(v) for

all u, v ∈ U and g(U) = U . If F acts as an antihomomorphism on U , then d = 0,

F is the identity map on N and N is a commutative ring.

Proof. First we show that d = 0 if and only if F is the identity map on N .

Clearly if F is the identity map on N , xd(y) = 0 for all x, y ∈ N , and hence

d = 0.

Conversely, assume that d = 0, in which case F (xy) = F (x)y = xF (y) for all

x, y ∈ N . It follows that for any x, y, z ∈ U ,

F (yxz) = F (z)F (yx) = F (z)yF (x) = F (zy)F (x) = zF (y)F (x) = zF (xy).

(3.2.1)

On the other hand,

F (yxz) = F (xz)F (y) = F (x)zF (y) = F (x)F (zy)

= F (x)F (y)F (z) = F (yx)F (z) = F (y)xF (z)

= F (y)F (xz) = F (y)F (x)z = F (xy)z. (3.2.2)

Comparing (3.2.1) and (3.2.2) shows that F (U2) centralizes U , so that F (U2) ⊆ Z

by Lemma 2.2.1(iii). Since U2 is a nonzero semigroup ideal; hence F (U2) 6= {0}

by Theorem 2.3.1. Suppose that x, y ∈ U such that F (xy) 6= 0, we see that for

any z ∈ U , F (xy)z = F (xyz) = F (yz)F (x) = F (y)F (zx) = F (y)F (x)F (z) =

F (xy)F (z), and hence F (xy)(z − F (z)) = 0. Since F (xy) ∈ Z\{0}, we get

F (z) = z for all z ∈ U . Hence F is the identity map on N . We note now

that if the identity map of N acts as an antihomomorphism on U , then U is

commutative, so that by Lemma 2.2.1(iii) and Lemma 2.2.3, N is a commutative

43

ring. To complete the proof of our theorem, we need only to argue that d = 0.

By our antihomomorphism hypothesis

F (xy) = d(x)g(y) + xF (y) = F (y)F (x) for all x, y ∈ U. (3.2.3)

Substituting xy for y in (3.2.3), we obtain

F (xy)F (x) = F (xxy) = d(x)g(xy) + xF (xy) for all x, y ∈ U.

F (xy)F (x) = d(x)g(xy) + xF (y)F (x).

(d(x)g(y) + xF (y))F (x) = d(x)g(xy) + xF (y)F (x).

Using Proposition 2.2.10(ii), we have

d(x)g(y)F (x) + xF (y)F (x) = d(x)g(x)g(y) + xF (y)F (x).

d(x)g(y)F (x) = d(x)g(x)g(y).

d(x)yF (x) = d(x)xy for all x, y ∈ U. (3.2.4)

Replacing y by yr in (3.2.4), we get

d(x)yrF (x) = d(x)xyr for all x, y ∈ U and r ∈ N.

Using (3.2.2), we have

d(x)yrF (x) = d(x)yF (x)r

and so,

d(x)y[r, F (x)] = 0 for all x, y ∈ U, r ∈ N.

Application of Lemma 2.2.1(i) yields that either d(x) = 0 or [r, F (x)] = 0 i.e.,

d(x) = 0 or F (x) ∈ Z. Suppose that there exists w ∈ U such that F (w) ∈ Z\{0}.

Then for all v ∈ U such that d(v) = 0, F (wv) = F (w)v + g(w)d(v) = F (w)v;

44

F (wv) = F (w)v = F (v)F (w) = F (w)F (v)

and hence F (w)(v − F (v)) = 0 = v − F (v).

Now consider arbitrary x, y ∈ U . If one of F (x), F (y) is in Z, then F (xy) =

F (x)F (y). If d(x) = 0 = d(y), then d(xy) = d(x)y + xd(y) = 0, so

F (xy) = xy = F (x)F (y). Therefore F (xy) = F (x)F (y) for all x, y ∈ U , and

by Theorem 3.2.2, F is the identity map on N , and therefore d = 0.

The remaining possibility is that for each x ∈ U , either d(x) = 0 or F (x) = 0.

Let u ∈ U\{0} and let U1 = uN . Then U1 is a nonzero semigroup right ideal

contained in U and U1 is an additive subgroup of N . The sets {x ∈ U1|d(x) = 0}

and {x ∈ U1|F (x) = 0} are additive subgroups of U1 with union equal to U1, so

d(U1) = {0} or F (U1) = {0}. If d(U1) = {0}, then d = 0 by Proposition 2.2.4.

Suppose, then, that F (U1) = {0}. Then for arbitrary x, y ∈ N , F (uxy) =

F (ux)y + g(ux)d(y) = 0 = uxd(y), so uNd(y) = {0}, and again d = 0. This

completes the proof.

3.3 Some commutativity conditions involvinggeneralized semiderivations

In this section we prove that a prime near ring N is a commutative ring if

the generalized semiderivation F : N −→ N satisfies one of the following:

(i) F ([u, v]) = [u, v]; (ii) F ([u, v]) = −[u, v]; (iii) F (u ◦ v) = 0; (iv) F ([u, v]) =

[F (u), v]; (v) F ([u, v]) = −[F (u), v]; (vi) F ([u, v]) = [u, F (v)]; (vii) F ([u, v]) =

−[u, F (v)] for all u, v ∈ U ; a nonzero semigroup ideal of N .


ideal of N . Suppose that N admits a nonzero generalized semiderivation F asso-

45

ciated with a nonzero semiderivation d and an additive map g associated with d

such that g(uv) = g(u)g(v) for all u, v ∈ U and g(U) = U . If F ([u, v]) = ±[u, v],

for all u, v ∈ U , then N is a commutative ring.

Proof. By hypothesis

F ([u, v]) = ±[u, v] for all u, v ∈ U. (3.3.1)

Replacing v by vu and using [u, vu] = [u, v]u, we get

F ([u, v]u) = ([u, v]u) for all u, v ∈ U.

Thus

F ([u, v])u+ g([u, v])d(u) = ±([u, v])u for all u, v ∈ U.

Using (3.3.1), we get

±([u, v])u+ g([u, v])d(u) = ±([u, v])u.

This implies that

g([u, v])d(u) = 0 for all u, v ∈ U.

Thus

g([u, v])d(U) = {0} for all u, v ∈ U,

and Proposition 2.2.6 yields that

g([u, v]) = 0 for all u, v ∈ U.

Therefore, we get

[g(u), g(v)] = 0 for all u, v ∈ U.

Since g(U) = U , we get

46

[u, v] = 0 for all u, v ∈ U. (3.3.2)

Now replacing u by ur, for all r ∈ N in (3.3.2), we find

U [r, v] = {0}.

Using Lemma 2.2.1(ii) we have [r, v] = 0 for all v ∈ U and r ∈ N . Hence U ⊆ Z

and N is commutative ring by Lemma 2.2.3.




such that g(uv) = g(u)g(v) for all u, v ∈ U and g(U) = U . If F (u ◦ v) = 0, for

all u, v ∈ U , then N is a commutative ring.

Proof. Assume that

F (u ◦ v) = 0 for all u, v ∈ U. (3.3.3)

Replacing v by vu in (3.3.3), we get

F (u ◦ vu) = 0 for all u, v ∈ U.

F (u(vu) + (vu)u) = 0

F ((uv + vu)u) = 0

F ((u ◦ v)u) = 0.

g(u ◦ v)d(u) = 0.

47

This implies that

g(u ◦ v)d(U) = {0} for all u, v ∈ U.

Using Proposition 2.2.6, we get

g(uv + vu) = 0 for all u, v ∈ U.

Since g is additive, we have

g(uv) + g(vu) = 0 for all u, v ∈ U.

g(u)g(v) + g(v)g(u) = 0 for all u, v ∈ U.

-i.e.,

g(u)g(v) = −g(v)g(u) for all u, v ∈ U. (3.3.4)

Replacing v by vw in (3.3.4), we get

g(u)g(v)g(w) = −g(v)g(w)g(u) for all u, v, w ∈ U.


g(v)g(u)g(w) = g(v)g(w)g(u) for all u, v, w ∈ U.

g(v)[g(u), g(w)] = 0 for all u, v, w ∈ U.

-i.e.,

U [g(u), g(w)] = {0} for all u,w ∈ U.

By Lemma 2.2.1(ii), [g(u), g(w)] = 0 for all u,w ∈ U . Since g(U) = U , it follows

that [u,w] = 0 for all u,w ∈ U . Arguing in the similar manner as in the proof of

Theorem 3.3.1, we get the result.

48




such that g(uv) = g(u)g(v) for all u, v ∈ U and g(U) = U . If F (u◦v) = ±(u◦v),

for all u, v ∈ U , then N is a commutative ring.

Proof. By hypothesis, we have

F (u ◦ v) = ±(u ◦ v), for all u, v ∈ U. (3.3.5)

Substituting vu for v in (3.3.5), we get

F (u ◦ vu) = ±(u ◦ vu).

F (u(vu) + (vu)u) = ±(u(vu) + vu(u)).

F ((uv + vu)u) = ±((uv + vu)u).

F ((u ◦ v)u) = ±((u ◦ v)u).

F (u ◦ v)u+ g(u ◦ v)d(u) = ±((u ◦ v)u).


±((u ◦ v)u) + g(u ◦ v)d(u) = ±((u ◦ v)u).

This implies that

g(u ◦ v)d(u) = 0.

Arguing in the similar manner as in the proof of Theorem 3.3.1 and Theorem

3.3.2, we get the required result.

49


semigroup ideal of N . If N admits a nonzero generalized semiderivation F as-

sociated with a nonzero semiderivation d and an additive map g associated with

d such that g(U) = U and g(uv) = g(u)g(v) for all u, v ∈ U , then the following

assertions are equivalent:

(i) F ([u, v]) = [F (u), v] for all u, v ∈ U .

(ii) F ([u, v]) = −[F (u), v] for all u, v ∈ U .

(iii) N is commutative ring.

Proof. It is obvious that (iii) implies both (i) and (ii). Now we prove that (i)

implies (iii). By hypothesis, we have

F ([u, v]) = [F (u), v] for all u, v ∈ U. (3.3.6)

Taking uv instead of v in (3.3.6) and noting that [u, uv] = u[u, v], we get

F ([u, uv]) = [F (u), uv].

F (u[u, v]) = [F (u), uv].

uF ([u, v]) + d(u)g([u, v]) = F (u)uv − uvF (u) for all u, v ∈ U.

Using (3.3.6) and recall that F (u)u = uF (u), we have

u[F (u), v] + d(u)g([u, v]) = F (u)uv − uvF (u).

uF (u)v − uvF (u) + d(u)g([u, v]) = F (u)uv − uvF (u).

F (u)uv − uvF (u) + d(u)g([u, v]) = F (u)uv − uvF (u).

50

d(u)g([u, v]) = 0 for all u, v ∈ U.

d(u)g(u)g(v) = d(u)g(v)g(u) for all u, v ∈ U. (3.3.7)

Replacing v by vt in (3.3.7), we get

d(u)g(u)g(vt) = d(u)g(vt)g(u), i.e.,

d(u)g(u)g(v)g(t) = d(u)g(v)g(t)g(u) for all u, v, t ∈ U.

Using (3.3.7), we find

d(u)g(v)g(u)g(t) = d(u)g(v)g(t)g(u) for all u, v, t ∈ U.

d(u)v(g(u)t− tg(u)) = 0

d(u)v[g(u), t] = 0 for all u, v, t ∈ U.

d(u)U [u, t] = {0} for all u, t ∈ U.

By Lemma 2.2.1(i) either d(U) = {0} or [u, t] = 0. If d(U) = {0}, we arrive at a

contradiction by Proposition 2.2.4. On the other hand if [u, t] = 0, then replacing

u by ur, we obtain

u[r, t] = 0 for all u, t ∈ U and r ∈ N.

Thus, U [r, t] = {0} and by Lemma 2.2.1(ii), we get

[r, t] = 0 for all t ∈ U and r ∈ N.

Hence U ⊆ Z and N is commutative ring by Lemma 2.2.3.

Now we prove that (ii) implies (iii). By hypothesis, we have

51

F ([u, v]) = −[F (u), v] for all u, v ∈ U. (3.3.8)

Taking uv instead of v in (3.3.8) and noting that [u, uv] = u[u, v], we get

F ([u, uv]) = −[F (u), uv].

F (u[u, v]) = −[F (u), uv].

uF ([u, v]) + d(u)g([u, v]) = −F (u)uv + uvF (u) for all u, v ∈ U.

Use (3.3.8) and recall that F (u)u = uF (u), to obtain

u(−[F (u), v]) + d(u)g([u, v]) = −F (u)uv + uvF (u).

−uF (u)v + uvF (u) + d(u)g([u, v]) = −F (u)uv + uvF (u).

−F (u)uv + uvF (u) + d(u)g([u, v]) = −F (u)uv + uvF (u).

d(u)g([u, v]) = 0 for all u, v ∈ U.

d(u)g(u)g(v) = d(u)g(v)g(u) for all u, v ∈ U. (3.3.9)

Replacing v by vt in (3.3.9), we get

d(u)g(u)g(vt) = d(u)g(vt)g(u), i.e.,

d(u)g(u)g(v)g(t) = d(u)g(v)g(t)g(u) for all u, v, t ∈ U.


d(u)g(v)g(u)g(t) = d(u)g(v)g(t)g(u) for all u, v, t ∈ U.

d(u)v(g(u)t− tg(u)) = 0

d(u)v[g(u), t] = 0 for all u, v, t ∈ U.

52

d(u)U [u, t] = {0} for all u, t ∈ U.

By Lemma 2.2.1(i) either d(U) = {0} or [u, t] = 0. If d(U) = {0}, we arrive at a

contradiction by Proposition 2.2.4. On the other hand if [u, t] = 0, then replacing

u by ur, we obtain

u[r, t] = 0 for all u, t ∈ U and r ∈ N.

Thus, U [r, t] = {0} and by Lemma 2.2.1(ii), we get

[r, t] = 0 for all t ∈ U and r ∈ N.

Hence U ⊆ Z and N is a commutative ring by Lemma 2.2.3.


semigroup ideal of N . If N admits a nonzero generalized semiderivation F as-

sociated with a nonzero semiderivation d and an additive map g associated with

d such that g(U) = U and g(uv) = g(u)g(v) for all u, v ∈ U , then the following

assertions are equivalent:

(i) F ([u, v]) = [u, F (v)] for all u, v ∈ U .

(ii) F ([u, v]) = −[u, F (v)] for all u, v ∈ U .

(iii) N is commutative ring.

Proof. It is obvious that (iii) implies both (i) and (ii). Now we prove that (i)

implies (iii). By hypothesis, we have

53

F ([u, v]) = [u, F (v)] for all u, v ∈ U. (3.3.10)

Replacing u by vu in (3.3.10), we get

F ([vu, v]) = [vu, F (v)] for all u, v ∈ U.

Using the fact that [vu, v] = v[u, v], we arrive at

F (v[u, v]) = [vu, F (v)], i.e.

vF ([u, v]) + d(v)g([u, v]) = vuF (v)− F (v)vu for all u, v ∈ U. (3.3.11)

Using (3.3.10) and noting that vF (v) = F (v)v, we have

v[u, F (v)] + d(v)g([u, v]) = vuF (v)− F (v)vu.

vuF (v)− vF (v)u+ d(v)g([u, v]) = vuF (v)− vF (v)u.

d(v)g([u, v]) = 0 for all u, v ∈ U.

Arguing in the similar manner as in the proof of Theorem 3.3.4, we get the result.

Now we prove that (ii) implies (iii). By hypothesis, we have

F ([u, v]) = −[u, F (v)] for all u, v ∈ U. (3.3.12)

Replacing u by vu in (3.3.12), we get

F ([vu, v]) = −[vu, F (v)] for all u, v ∈ U.

Using the fact that [vu, v] = v[u, v], we arrive at

F (v[u, v]) = −[vu, F (v)], i.e.

54

vF ([u, v]) + d(v)g([u, v]) = −vuF (v) + F (v)vu for all u, v ∈ U. (3.3.13)

Using (3.3.12) and noting that vF (v) = F (v)v, we have

v(−[u, F (v)]) + d(v)g([u, v]) = −vuF (v) + F (v)vu.

−vuF (v) + vF (v)u+ d(v)g([u, v]) = −vuF (v) + vF (v)u.

d(v)g([u, v]) = 0 for all u, v ∈ U.

Using similar arguments as we have used in the proof of Theorem 3.3.4, we get

the required result.

The following example justifies the fact that the above theorems do not hold

for arbitrary near rings.

Example 3.3.1 Let S be a 2-torsion free left near ring and let

N =

{ 0 0 x0 0 y0 0 0

| x, y ∈ S} .

Define F, d, g : N → N by

F

0 0 x0 0 y0 0 0

=

0 0 00 0 y0 0 0

; d

0 0 x0 0 y0 0 0

=

0 0 x0 0 00 0 0

and

g

0 0 x0 0 y0 0 0

=

0 0 y0 0 x0 0 0

respectively. It can be easily checked that N is a left near ring and F is a

generalized semiderivation of N associated with a semiderivation d and onto map

55

g associated with d satisfying:

(i) F ([A,B]) = ±[A,B]; (ii) F (A ◦B) = 0; (iii) F ([A,B]) = ±[F (A), B];

(iv) F ([A,B]) = ±[A,F (B)]; for all A,B ∈ N . However N is not commutative

ring.

56

Chapter 4

Functional identities with pair ofgeneralized Semiderivations

4.1 Introduction

In 1957 E. C. Posner [166] established two very striking results (i) if R is

a prime ring admitting a nonzero derivation d such that [x, d(x)] = 0 for all

x ∈ R, then R must be commutative and (ii) if R is a prime ring of characteristic

different from two and d1, d2 are derivations such that the iterate d1d2 is also a

derivation, then at least one of d1, d2 is zero. A number of researchers generalized

these results in various directions to mention a few: Beidar, Bresar, Mayne, Her-

stein, Deng, Meirs, Bergen, Chebotar, Chuang, Hirano, Jensen, Krempa, Lanski,

Vukman, Lee, Bell, Martindale.

In section 4.2 we prove some results that are necessary to develop the proof

of our main theorems.

In section 4.3 we obtain some generalizations/extensions of Posner’s theo-

rems for a generalized semiderivation of a prime near ring.

Section 4.4 is devoted to the study of functional identities involving pair of

generalized semiderivations of a near ring. We prove that a prime near ringN with

generalized semiderivations F1 and F2 satisfying F1(x)F2(y)+F2(y)F1(x) ∈ Z for

57

all x, y ∈ U , a nonzero semigroup ideal is a commutative ring under some condi-

tions.

4.2 Preliminary results

Lemma 4.2.1 [44, Lemma 2.4] Let N be an arbitrary near ring. Let S and T

be non empty subsets of N such that st = −ts for all s ∈ S and t ∈ T . If a, b ∈ S

and c is an element of T for which −c ∈ T , then (ab)c = c(ab).

Now we prove the following Propositions which extend the results of [14, Lemmas

5-8], [44, Theorem 3.1 and Lemma 1.7] and [43, Lemma 2.2].

Proposition 4.2.2 Let N be a 3-prime near ring admitting a generalized

semiderivation F associated with a semiderivation d and an additive map g asso-

ciated with d. Then N satisfies the following laws:

(i) d(x)y + g(x)d(y) = g(x)d(y) + d(x)y for all x, y ∈ N.

(ii) d(x)g(y) + xd(y) = xd(y) + d(x)g(y) for all x, y ∈ N.

(iii) F (x)y + g(x)d(y) = g(x)d(y) + F (x)y for all x, y ∈ N.

(iv) d(x)g(y) + xF (y) = xF (y) + d(x)g(y) for all x, y ∈ N.

Proof. (i) d(x(y+y)) = d(x)(y+y)+g(x)d(y+y) = d(x)y+d(x)y+g(x)d(y)+

g(x)d(y), and d(xy+xy) = d(xy) + d(xy) = d(x)y+ g(x)d(y) + d(x)y+ g(x)d(y).

Comparing these two equations, we obtain the result.

(ii) d((x+x)y) = d(x+x)y+g(x+x)d(y) = d(x)y+d(x)y+g(x)d(y)+g(x)d(y) and

58

d(xy + xy) = d(xy) + d(xy) = d(x)y + g(x)d(y) + d(x)y + g(x)d(y). Comparing

these two equations, we get the result.

(iii) F (x(y+y)) = F (x)(y+y)+g(x)d(y+y) = F (x)y+F (x)y+g(x)d(y)+g(x)d(y),

and F (xy + xy) = F (x)y + g(x)d(y) + F (x)y + g(x)d(y). Comparing these two

equations, get the result.

(iv) F ((x+x)y) = F (x+x)y+g(x+x)d(y) = F (x)y+F (x)y+g(x)d(y)+g(x)d(y),

and F (xy + xy) = F (x)y + g(x)d(y) + F (x)y + g(x)d(y). Comparing these two

equations, we get the desired result.


ideal of N . Suppose that N admits nonzero semiderivations d1, d2 associated with

a map g such that g(uv) = g(u)g(v) for all u, v ∈ U . If d1(x)d2(y)+d2(y)d1(x) ∈ Z

for all x, y ∈ U and at least one of d1(U) ∩ Z and d2(U) ∩ Z is nonzero, then N

is a commutative ring.

Proof. Assume that d1(U) ∩ Z 6= {0}. Let x ∈ U such that d1(x) ∈ Z\{0}

and y ∈ U . Then d1(x)d2(y) + d2(y)d1(x) = d1(x)(2d2(y)) = d1(x)(d2(2y)) ∈ Z.

Therefore, d2(2U) ⊆ Z. Since 2U is nonzero semigroup left ideal, our conclusion

follows by Proposition 2.2.8, then N is commutative ring.


nonzero semigroup ideal of N . Then 2U 6= {0} and d(2U) 6= {0} for any nonzero

semiderivation d associated with a map g such that g(U) = U .

Proof. Let x ∈ N with x+x 6= 0. Then for every u ∈ U , u(x+x) = ux+ux ∈ 2U ;

59

and by Lemma 2.2.1(ii), we get {0} 6= U(x + x) ⊆ 2U . Since 2U is a semigroup

left ideal, it follows by Proposition 2.2.4 that d(2U) 6= {0}.

Proposition 4.2.5 Let N be a 3-prime near ring. If F is a generalized

semiderivation with associated semiderivation d and a map g associated with

d such that g(U) = U , then F (Z) ⊆ Z.

Proof. Using similar techniques as [14, Lemma 7]. Let z ∈ Z and x ∈ N . Then

F(zx)=F(xz); that is F (z)x + g(z)d(x) = d(x)g(z) + xF (z). Applying Proposi-

tion 4.2.2 (iii), we get g(z)d(x) + F (z)x = d(x)g(z) + xF (z); zd(x) + F (z)x =

d(x)z + xF (z). It follows that F (z)x = xF (z) for all x ∈ N , so F (Z) ⊆ Z.


ideal of N . Suppose that N admits a semiderivation d associated with a map g

such that g(U) = U and g(uv) = g(u)g(v) for all u, v ∈ U . If d2(U) 6= {0} and

a ∈ N such that [a, d(U)] = {0}, then a ∈ Z.

Proof. Let C(a) = {x ∈ N |ax = xa}. Note that d(U) ⊆ C(a). Thus, if

y ∈ C(a) and u ∈ U , both d(yu) and d(u) are in C(a); hence (d(y)g(u)+yd(u))a =

a(d(y)g(u) + yd(u)) and using Proposition 2.2.5, we get d(y)g(u)a + yd(u)a =

ad(y)g(u) + ayd(u); d(y)ua+ yd(u)a = ad(y)u+ ayd(u). Since yd(u) ∈ C(a), we

conclude that d(y)ua = ad(y)u. Thus

d(C(a))U ⊆ C(a). (4.2.1)

Choosing z ∈ U such that d2(z) 6= 0, and let y = d(z). Then y ∈ C(a); and

by (4.2.1), d(y)u ∈ C(a) and d(y)uv ∈ C(a) for all u, v ∈ U . Therefore,

0 = [a, d(y)uv] = ad(y)uv − d(y)uva = d(y)uav − d(y)uva = d(y)u(av − va).

60

Thus d(y)U(av − va) = 0 for all v ∈ U ; and by Lemma 2.2.1(i), a centralizes U .

Another appeal to Lemma 2.2.1(iii) yields that a ∈ Z.

Proposition 4.2.7 Let N be a 3-prime near ring and F be a generalized

semiderivation of N with associated nonzero semiderivation d and a map g as-

sociated with d such that g is onto and g(xy) = g(x)g(y) for all x, y ∈ N .

If d(F (N)) = {0}, then d2(x)d(y) + d(x)d2(y) = 0 for all x, y ∈ N and

F (d(N)) = {0}.

Proof. We are assuming that d(F (x)) = 0 for all x ∈ N . It follows that

d(F (xy)) = d(F (x)y) + d(g(x)d(y)) = d(F (x)y) + d(xd(y)) = 0 for all x, y ∈ N ,

that is,

d(F (x))g(y) + F (x)d(y) + d(x)g(d(y)) + xd2(y) = 0 for all x, y ∈ N.

This implies that

F (x)d(y) + d(x)d(g(y)) + xd2(y) = 0.

F (x)d(y) + d(x)d(y) + xd2(y) = 0 for all x, y ∈ N. (4.2.2)

Applying d again, we get

F (x)d2(y) + d2(x)d(y) + d(x)d2(y) + d(x)d2(y) + xd3(y) = 0 for all x, y ∈ N.

(4.2.3)

Taking d(y) instead of y in (4.2.2) gives F (x)d2(y)+d(x)d2(y)+xd3(y) = 0, hence

(4.2.3) yields that

d2(x)d(y) + d(x)d2(y) = 0 for all x, y ∈ N. (4.2.4)

Now, substitute d(x) for x in (4.2.3), obtaining

F (d(x))d(y) + d2(x)d(y) + d(x)d2(y) = 0;

61

and use (4.2.4) to conclude that F (d(x))d(y) = 0 for all x, y ∈ N . Thus, by

Theorem 2.3.2, F (d(x)) = 0 for all x ∈ N .

4.3 Extension of Posner’s Theorems

Posner’s first theorem states that if R is a prime ring with a nonzero deriva-

tion d such that [x, d(x)] = 0 for all x ∈ R, then R must be commutative. A

number of authors generalized/extended this theorem in many ways for example

see [32], [49], [70], [71], [72], [73], [87], [146], [147], [148] and [152] where further

references can be found. In [34] Beidar, Fong and Wang proved that if N is a

3-prime near ring and d : N −→ N is a nonzero derivation such that d2 6= 0 and

for a ∈ N , [a, d(x)] = 0 for all x ∈ N , then a ∈ Z, the centre of N . Further Bell

[41] extended the result for a semigroup ideal of N . In this section we obtain

the result for a generalized semiderivation F of N . More precisely we prove the

following:

Theorem 4.3.1 Let N be a 2-torsion free 3-prime near ring and F be a nonzero

generalized semiderivation of N with associated semiderivation d and a map g

associated with d such that g(U) = U ; g(uv) = g(u)g(v) for all u, v ∈ U and

F (V ) ⊆ U for some nonzero semigroup ideal V contained in U . If a ∈ N and

[a, F (U)] = {0}, then a ∈ Z.

Proof. If d = 0, then for all x ∈ U and y ∈ N , aF (x)y = F (x)ya; hence

F (U)[a, y] = {0} and a ∈ Z by Theorem 2.3.2. Therefore, we may assume

d 6= 0. Let C(a) denotes the centralizer of a, and let y ∈ C(a) for all u ∈ U ,

F (uy) ∈ C(a) -i.e. (F (u)y+g(u)d(y))a = a(F (u)y+g(u)d(y)) and by Proposition

2.2.10(i), F (u)ya+g(u)d(y)a = aF (u)y+ag(u)d(y); F (u)ya+ud(y)a = aF (u)y+

62

aud(y). Now F (u)ya = aF (u)y, and it follows that d(y)u ∈ C(a); therefore

d(C(a))U is a semigroup right ideal which centralizes a, and if d(C(a))U 6= {0}.

Lemma 2.2.1(iii) yields a ∈ Z. Assume now that d(C(a))U = {0}, in which

case d(C(a)) = {0} and hence d(F (U)) = {0}. It follows that for all x ∈ N

and v ∈ V , d(F (xF (v))) = 0 = d(F (x)F (v) + g(x)d(F (v))) = d(F (x)F (v)) =

d(F (x))g(F (v)) +F (x)d(F (v)) = d(F (x))F (v), so that d(F (N))F (V ) = {0} and

by Theorem 2.3.2, d(F (N)) = {0}. By Proposition 4.2.7

d2(x)d(y) + d(x)d2(y) = 0 for all x, y ∈ N and F (d(N)) = {0}. (4.3.1)

As in the proof of Theorem 4.1 of [43], we calculate F (d(x)d(y)) in two ways,

obtaining F (d(x)d(y)) = F (d(x))d(y) + g(d(x))d2(y) = d(g(x))d2(y) = d(x)d2(y)

and F (d(x)d(y)) = d2(x)g(d(y))+d(x)F (d(y)) = d2(x)d(g(y)) = d2(x)d(y). Com-

paring the two results, we get d(x)d2(y) = d2(x)d(y) for all x, y ∈ N , which

together with (4.3.1) gives d2(x)d(y) = 0 for all x, y ∈ N and hence d2 = 0.

But by Proposition 2.2.7, this contradicts our assumption that d 6= 0; thus

d(C(a))U 6= {0} and our proof is complete.

Posner’s second theorem states as follows:

Theorem 4.3.2 [166, Theorem 1] If R is a prime ring of characteristic differ-

ent from two and d1, d2 are derivations of R such that the iterate d1d2 is also a

derivation, then at least one of d1, d2 is zero.

Thereafter a number of authors have extended/generalized the theorem in

several directions for example Bergen [54], Chebotar [79], Chuang [82], Hi-

rano et.al. [113], Hvala [116], Jensen [118], Krempa [124], Martindale [143],

63

Ye et.al. [190]. In 1994 Wang [189] investigated the following result in case of

near rings which is one of the extensions of Theorem 4.3.2.

Theorem 4.3.3 Let N be a 2-torsion free prime near ring and d1, d2 be deriva-

tions of N such that d1d2 is also a derivation. Then the following conditions are

equivalent:

(i) Either d1 = 0 or d2 = 0.

(ii) [d1(x), d2(y)] = 0 for all x, y ∈ N .

Beidar et.al. [34] obtained the following result:

Theorem 4.3.4 Let N be a prime near ring with derivations d1 and d2 such that

d1(x)d2(y) = −d2(x)d1(x) for all x, y ∈ N . Suppose that 2N 6= {0}. Then either

d1 = 0 or d2 = 0.

Further Bell and Argac [44] extended these results for a semigroup ideal of a

3-prime near ring. In this line of investigations we prove the following theorems:


semigroup ideal of N . Let F1 and F2 be generalized semiderivations on N with

associated semiderivations d1 and d2 with at least one of d1, d2 not zero and a

map g associated with d1 and d2 such that g(uv) = g(u)g(v) for all u, v ∈ U and

g(U) = U . If F1(x)d2(y)+F2(x)d1(y) = 0 for all x, y ∈ U , then F1 = 0 or F2 = 0.

Proof. By hypothesis

F1(x)d2(y) + F2(x)d1(y) = 0 for all x, y ∈ U. (4.3.2)

64

Replacing x by uv in (4.3.2), we get

F1(uv)d2(y) + F2(uv)d1(y) = 0 for all x, y ∈ U.

(d1(u)g(v) + uF1(v))d2(y) + (d2(u)g(v) + uF2(v))d1(y) = 0 for all u, v, y ∈ U.

Using Proposition 2.2.10(ii) and Proposition 4.2.2(iv), we conclude that

(d1(u)g(v) + uF1(v))d2(y) + (uF2(v) + d2(u)g(v))d1(y) = 0.

d1(u)g(v)d2(y) + uF1(v)d2(y) + uF2(v)d1(y) + d2(u)g(v)d1(y) = 0.

d1(u)vd2(y) + u(F1(v)d2(y) + F2(v)d1(y)) + d2(u)vd1(y) = 0 for all u, v, y ∈ U.

Since middle summand is 0 by (4.3.2), we conclude that

d1(u)vd2(y) + d2(u)vd1(y) = 0 for all u, v, y ∈ U. (4.3.3)

Substituting yt for y in (4.3.3), we get

d1(u)vd2(yt) + d2(u)vd1(yt) = 0 for all u, v, y, t ∈ U.

d1(u)v(d2(y)g(t) + yd2(t)) + d2(u)v(d1(y)g(t) + yd1(t)) = 0.

Using Proposition 4.2.2(ii), we have

d1(u)v(d2(y)g(t) + yd2(t)) + d2(u)v(yd1(t) + d1(y)g(t)) = 0.

This implies that

d1(u)vd2(y)t+ (d1(u)vyd2(t) + d2(u)vyd1(t)) + d2(u)vd1(y)t = 0.

Again the middle summand is 0, so

d1(u)vd2(y)t+ d2(u)vd1(y)t = 0 for all u, v, y, t ∈ U. (4.3.4)

65

Replacing t by td1(w) in (4.3.4), where w ∈ U , we have

d1(u)v(d2(y)td1(w)) + d2(u)(vd1(y)t)d1(w) = 0 for all u, v, y, t, w ∈ U.


d1(u)v(−d1(y)td2(w))− d1(u)vd1(y)td2(w) = 0.

This implies that

2d1(u)vd1(y)td2(w) = 0 for all u, v, y, t, w ∈ U.

Since N is 2-torsion free, we get

d1(u)vd1(y)td2(w) = 0 for all u, v, y, t, w ∈ U.

Thus d1(U)Ud1(U)Ud2(U) = {0}; and by Lemma 2.2.1(i) and Proposition 2.2.4,

one of d1, d2 must be 0. Assuming without loss that d1 = 0, in which case d2 6= 0,

we get F1(U)d2(U) = {0}, so by Proposition 2.2.6 and Theorem 2.3.1, we have

F1 = 0.



associated semiderivations d1 and d2 and a map g associated with d1 and d2 such

that g(uv) = g(u)g(v) for all u, v ∈ U and g(U) = U . If d1 and d2 are not

both zero and F1F2 acts on U as a generalized semiderivation with associated

semiderivation d1d2 and a map g associated with d1d2, then F1 = 0 or F2 = 0.

Proof. By the hypothesis, we have

F1F2(xy) = F1F2(x)y + g(x)d1d2(y) for all x, y ∈ U.

66

F1F2(xy) = F1F2(x)y + xd1d2(y) for all x, y ∈ U. (4.3.5)

We also have

F1F2(xy) = F1(F2(xy)) = F1(F2(x)y + g(x)d2(y))

= F1(F2(x)y) + F1(g(x)d2(y))

= F1(F2(x)y) + F1(xd2(y)).

-i.e.,

F1F2(xy) = F1F2(x)y + g(F2(x))d1(y) + F1(x)d2(y) + g(x)d1d2(y)

= F1F2(x)y + F2(g(x))d1(y) + F1(x)d2(y) + g(x)d1d2(y)

= F1F2(x)y+F2(x)d1(y) +F1(x)d2(y) +xd1d2(y) for all x, y ∈ U. (4.3.6)

Comparing (4.3.5) and (4.3.6), we get

F2(x)d1(y) + F1(x)d2(y) = 0 for all x, y ∈ U.

Hence application of Theorem 4.3.5 yields that F1 = 0 or F2 = 0.




that g(uv) = g(u)g(v) for all u, v ∈ U and g(U) = U . If F1F2(U) = {0}, then

F1 = 0 or F2 = 0.

Proof. By the hypothesis

F1F2(U) = {0}.

F1F2(xy) = F1(F2(xy)) = 0 = F1(F2(x)y + g(x)d2(y))

67

= F1(F2(x)y) + F1(xd2(y))

= F1F2(x)y + g(F2(x))d1(y) + F1(x)d2(y) + g(x)d1d2(y)

= F2(g(x))d1(y) + F1(x)d2(y) + xd1d2(y) for all x, y ∈ U.

This implies that

F2(x)d1(y) + xd1d2(y) + F1(x)d2(y) = 0 for all x, y ∈ U. (4.3.7)

Replacing x by zx in (4.3.7), we have

F2(zx)d1(y) + zxd1d2(y) + F1(zx)d2(y) = 0 for all x, y, z ∈ U.

(d2(z)g(x) + zF2(x))d1(y) + zxd1d2(y) + (d1(z)g(x) + zF1(x))d2(y) = 0.

Using Proposition 2.2.10(ii) and Proposition 4.2.2(iv), we get

(d2(z)g(x) + zF2(x))d1(y) + zxd1d2(y) + (zF1(x) + d1(z)g(x))d2(y) = 0.

d2(z)g(x)d1(y) + zF2(x)d1(y) + zxd1d2(y) + zF1(x)d2(y) + d1(z)g(x)d2(y) = 0.

d2(z)xd1(y) + z(F2(x)d1(y) + xd1d2(y) + F1(x)d2(y)) + d1(z)xd2(y) = 0.

Since the middle summand is 0 by (4.3.7), we have

d2(z)xd1(y) + d1(z)xd2(y) = 0 for all x, y, z ∈ U.

But this is just (4.3.3) of Theorem 4.3.5, so we argue as in the proof of Theorem

4.3.5 that d1 = 0 or d2 = 0. It now follows from (4.3.7) that

F2(x)d1(y) + F1(x)d2(y) = 0 for all x, y ∈ U.

If one of d1, d2 is nonzero, then F1 or F2 is 0 by Theorem 4.3.5, so we assume that

d1 = d2 = 0. Then F1F2(xy) = 0 = F1(F2(x)y) = F2(x)F1(y) for all x, y ∈ U , so

68

that F2(U)F1(U) = {0}. Applying Theorem 2.3.2, we conclude that F1 = 0 or

F2 = 0.

We now consider a somewhat different condition that elements of F1(U) and

F2(U) anti-commute.

Theorem 4.3.8 Let N be a 2-torsion free 3-prime near ring with nonzero semi-

group ideal U ; and let F1 and F2 be generalized semiderivations on N with asso-

ciated semiderivations d1 and d2 such that F1(U2) ⊆ U and F2(U

2) ⊆ U and a

map g associated with d1 and d2 such that g(uv) = g(u)g(v) for all u, v ∈ U and

g(U) = U . If

F1(x)F2(y) + F2(y)F1(x) = 0 for all x, y ∈ U, (4.3.8)

then F1 = 0 or F2 = 0.

Proof. Assume that F1 6= 0 and F2 6= 0. Note that if w ∈ F2(U2), −w ∈ F2(U);

and apply Lemma 4.2.1 to get (uv)w = w(uv) for all u, v ∈ F1(U) and w ∈ F2(U2).

It follows by Theorem 4.3.1 that F1(U)F1(U) ⊆ Z, and it is easy to see that

F1(x)F1(y)(F1(x)F1(y)− F1(y)F1(x)) = 0 for all x, y ∈ U.

This implies that

F1(y)F1(x)(F1(x)F1(y)− F1(y)F1(x)) = 0 for all x, y ∈ U.

Since F1(x)F1(y) and F1(y)F1(x) are central. Lemma 2.2.2(i) shows that either

both are zero or one can be cancelled to yield

F1(x)F1(y) = F1(y)F1(x).

69

Thus [F (U), F1(U)] = {0} and by Theorem 4.3.1, F1(U) ⊆ Z, hence N is

a commutative ring by Theorem 2.4.2. This fact together with (4.3.8) gives

F1(U)F2(U) = {0}. Contradicting our assumption that F1 6= 0 6= F2. Therefore

F1 = 0 or F2 = 0 as required.

If U is closed under addition, then F (U2) ⊆ U for any generalized semideriva-

tion F ; hence we have

Corollary 4.3.9 Let N be a 2-torsion free 3-prime near ring and U be a nonzero

semigroup ideal of N which is closed under addition. If F1 and F2 are generalized

semiderivations on N with associated semiderivations d1 and d2 and a map g

associated with d1 and d2 such that g(uv) = g(u)g(v) for all u, v ∈ U and g(U) =

U ; such that

F1(x)F2(y) + F2(y)F1(x) = 0 for all x, y ∈ U,

then F1 = 0 or F2 = 0.

We now replace the hypothesis that F1(U) ⊆ U and F2(U) ⊆ U in Theorem

4.3.8 by some commutativity hypothesis.

Theorem 4.3.10 Let N be a 2-torsion free 3-prime near ring with nonzero

semigroup ideal U ; and let F1 and F2 be generalized semiderivations on N with


that g(U) = U and g(uv) = g(u)g(v) for all u, v ∈ U ; such that

F1(x)F2(y) + F2(y)F1(x) = 0 for all x, y ∈ U.

Then F1 = 0 or F2 = 0 if one of the following is satisfied: (a) d1(Z) 6= {0} and

d2(Z) 6= {0}; (b) U ∩ Z 6= {0}.

70

Proof. (a) Let z1 ∈ Z such that d1(z1) 6= 0. Then for all x, y ∈ U , we have

F1(z1x)F2(y) + F2(y)F1(z1x) = 0.

(d1(z1)g(x) + z1F1(x))F2(y) + F2(y)(F1(x)z1 + g(x)d1(z1)) = 0.

d1(z1)g(x)F2(y) + z1F1(x)F2(y) + F2(y)F1(x)z1 + F2(y)g(x)d1(z1) = 0.

d1(z1)xF2(y) + z1(F1(x)F2(y) + F2(y)F1(x)) + F2(y)xd1(z1) = 0.

It follows that

d1(z1)xF2(y) + F2(y)xd1(z1) = 0 for all x, y ∈ U.

Choosing z2 ∈ Z such that d2(z2) 6= 0 and using a similar argument, we now get

xy + yx = 0 for all x, y ∈ U ;

and applying Lemma 4.2.1 with S = U and T = U2 shows that U2 centralizes

U2, so that U2 ⊆ Z by Lemma 2.2.1(iii) and hence N is commutative ring by

Lemma 2.2.3. It now follows that F1(x)F2(y) = F2(y)F1(x) = −F2(y)F1(x) for

all x, y ∈ U . Hence F1(U)F2(U) = {0}. Therefore F1 = 0 or F2 = 0.

(b) We assume that F1 6= 0 and F2 6= 0. Let z0 ∈ (U ∩ Z)\{0}. By Proposition

4.2.5, F1(z0) ∈ Z; hence if F1(z0) 6= 0 the condition

F1(z0)F2(x) + F2(x)F1(z0) = 0 for all x ∈ U

gives 2F2(x) = 0 = F2(x) for all x ∈ U , so that F1 = 0 by Theorem 2.3.1.

Therefore, F1(z0) = 0 and similarly F2(z0) = 0. Now z20 ∈ (U ∩ Z)\{0} also, so

F1(z20) = 0 = F2(z

20); and since F1(z

20) = F1(z0)z0 + g(z0)d1(z0) = z0d1(z0) and

71

F2(z20) = F2(z0)z0 + g(z0)d2(z0) = z0d2(z0). we have d1(z0) = d2(z0) = 0. Ob-

serving that F1(z0x) = F1(z0)x + g(z0)d1(x) = F1(z0)x + z0d1(x) and F1(xz0) =

F1(x)z0 +g(x)d1(z0) = F1(x)z0 +xd1(z0) for all x ∈ N , we see that F1(x) = d1(x)

for all x ∈ N , So that F1 is a semiderivation; and similarly F2 is a semideriva-

tion. We can now derive a contradiction as in the proof of Theorem 4.3.8, with

Proposition 2.2.7 and Proposition 4.2.6 used instead of Theorem 4.3.1.

4.4 Product of generalized semiderivations andcommutativity of prime near rings

The skew-commutativity hypothesis of Theorem 4.3.7 and 4.3.8 suggests

investigating conditions of the form F1(x)F2(y) + F2(y)F1(x) ∈ Z or xF (y) +

F (y)x ∈ Z.


semigroup ideal of N which is closed under addition.

(i) If N has nonzero generalized semiderivations F1, F2 with associated

semiderivations d1 and d2 and a map g associated with d1 and d2 such that g(U) =

U and g(uv) = g(u)g(v) for all u, v ∈ U ; such that F1(x)F2(y) +F2(y)F1(x) ∈ Z,

for all x, y ∈ U and at least one of F1(U)∩Z and F2(U)∩Z is nonzero, then N

is a commutative ring.

(ii) If N admits a nonzero generalized semiderivation F with associated

semiderivation d and a map g associated with d such that g(U) = U and

g(uv) = g(u)g(v) for all u, v ∈ U ; such that U ∩Z 6= {0} and xF (y)+F (y)x ∈ Z,

for all x, y ∈ U , then N is commutative ring.

72

Proof. (i) Assume that F1(U) ∩ Z 6= {0}. Let x ∈ U such that F1(x) ∈ Z\{0}.

Then F1(x)F2(y) + F2(y)F1(x) = 2F1(x)F2(y) = F1(x)F2(2y) ∈ Z for all y ∈ U .

Since F1(x) ∈ Z\{0}, Lemma 2.2.2(iii) gives F2(2y) ∈ Z for all y ∈ U -i.e.

F2(2U) ⊆ Z. Since 0 ∈ Z, we get F2(2U) = {0} -i.e. 2F2(U) = {0}. But N is

2-torsion free, we get F2(U) = {0} would contradict our hypothesis that F2 6= 0;

hence F2(2U) 6= {0} and we may choose y ∈ U such that F2(2y) ∈ Z\{0}. Since

2U ⊆ U , this shows that F2(2y) and 2F2(2y) = F2(4y) are in F2(U) ∩ Z\{0}, so

that for all x ∈ U , F1(x)(2F2(2y)) ∈ Z and hence F1(x) ∈ Z. Thus, F1(U) ⊆ Z

and by Theorem 2.4.1, N is a commutative ring.

(ii) Assume that (U) ∩ Z 6= {0}. Let x ∈ U such that x ∈ Z\{0}. Then

xF2(y) + F2(y)x = 2xF2(y) = xF2(2y) ∈ Z for all y ∈ U . Since x ∈ Z\{0},

Lemma 2.2.2(iii) gives F2(2y) ∈ Z for all y ∈ U i.e. F2(2U) ⊆ Z. Since 0 ∈ Z,

we get F2(2U) = {0} -i.e., 2F2(U) = {0}. But N is 2-torsion free, we get

F2(U) = {0}. By Theorem 2.3.1, we get F2 = 0 – a contradict our hypoth-

esis that F2 6= 0; hence F2(2U) 6= {0} and we may choose y ∈ U such that

F2(2y) ∈ Z\{0}. Since for any u ∈ U , we get ur ∈ U for r ∈ N . Replacing r

by r + s, for r, s ∈ N we get u(r + s) ∈ U -i.e., ur + us ∈ U or u′ + v′ ∈ U

for all u′, v′ ∈ U . Which yields that U + U ⊆ U -i.e., 2U ⊆ U , this shows that

F2(2y) and 2F2(2y) = F2(4y) are in F2(U) ∩ (Z \ {0}), so that for all x ∈ U ,

x(2F2(2y)) ∈ Z and hence x ∈ Z. Thus, U ⊆ Z and by Theorem 2.4.1, N is a

commutative ring.

The following theorem is an extension of Theorem 3.2 of [44].

73


semigroup ideal of N which is closed under addition. Suppose N admits nonzero

generalized semiderivations F1 and F2 with associated semiderivations d1 and d2

and a map g associated with d1 and d2 such that g(U) = U and g(uv) = g(u)g(v)

for all u, v ∈ U ; such that F1(x)F2(y) + F2(y)F1(x) ∈ Z, for all x, y ∈ U and

F1(U) ⊆ U and F2(U) ⊆ U . If F1(N) ∩ Z 6= {0} or F2(N) ∩ Z 6= {0}, then N is

a commutative ring.

Proof. By corollary 4.3.9, we cannot have F1(x)F2(y) + F2(y)F1(x) = 0 for

all x, y ∈ U , hence there exist x0, y0 ∈ U such that u0 = F1(x0)F2(y0) +

F2(y0)F1(x0) ∈ (Z\{0}) ∩ U . Since F1(Z) and F2(Z) are central by Proposition

4.2.5, if F1(u0) 6= 0 or F2(u0) 6= 0 we have F1(U) ∩ Z 6= {0} or F2(U) ∩ Z 6= {0}

and our conclusion follows by Theorem 4.4.1 (i).

Assume, therefore, that F1(u0) = F2(u0) = 0. For all x, y ∈ U , F1(u0x)F2(u0y) +

F2(u0y)F1(u0x) = u20(d1(x)d2(y) + d2(y)d1(x)) ∈ Z, hence d1(x)d2(y) +

d2(y)d1(x) ∈ Z; and if d1(u0) 6= 0 or d2(u0) 6= 0 our desired conclusion follows by

Proposition 4.2.3. Therefore we may assume d1(u0) = d2(u0) = 0. For all x, y ∈

N , F1(xu0)F2(yu0) + F2(yu0)F1(xu0) ∈ Z, so u20(F1(x)F2(y) + F2(y)F1(x)) ∈ Z

and F1(x)F2(y) + F2(y)F1(x) ∈ Z. Since F1(N) ∩ Z 6= {0} or F2(N) ∩ Z 6= {0},

our result follows by Theorem 4.4.1 (i).

74

Chapter 5

Functional identities with tracesof generalized n-derivations

5.1 Introduction

Ozturk et.al. [159] and Park et.al. [163] studied bi-derivations and tri-

derivations in near rings. A mapping D : N ×N −→ N is said to be symmetric

(permuting) on a near ring N if D(x, y) = D(y, x) for all x, y ∈ N . A symmetric

bi-additive mapping D : N ×N −→ N (additive in each argument) is said to be

a symmetric bi-derivation on N if D(xy, z) = D(x, z)y + xD(y, z) holds for all

x, y, z ∈ N . A permuting tri-additive mapping ∆ : N ×N ×N −→ N is said to

be a permuting tri-derivation on N if

∆(xw, y, z) = D(x, y, z)w + xD(w, y, z)

is fulfilled for all w, x, y, z ∈ N . Very recently Park [162] defined n-derivations

in ring. Analogously we define n-derivations in near ring. Let n ≥ 2 be a fixed

positive integer and Nn = N ×N × ....×N︸︷︷︸n−times

. An n-additive map ∆ : Nn −→ N

(additive in each argument) is said to be an n-derivation on a near ring N if the

relations

∆(x1x′1, x2, ...., xn) = ∆(x1, x2, ...., xn)x′1 + x1∆(x′1, x2, ...., xn)

∆(x1, x2x′2, ...., xn) = ∆(x1, x2, ...., xn)x′2 + x2∆(x1, x

′2, ...., xn)

:

∆(x1, x2, ...., xnx′n) = ∆(x1, x2, ...., xn)x′n + xn∆(x1, x2, ...., x

′n)

75

hold for all x1, x′1, x2, x

′2, ....., xn, x

′n ∈ N . If ∆ is a permuting map, then ∆ is

called a permuting n-derivation on N .

In case of rings and near rings derivations and bi-derivations have received

significant attention in recent years see [41], [77], [159], [161], [166], [183], [187]

and [188]. In [164] Park and Jung studied generalized tri-derivations in near

rings. A permuting 3-additive map F : N × N × N −→ N (additive in

each argument) on a near ring N is said to be a permuting left generalized

3-derivation (resp. permuting right generalized 3-derivation) associated with

3-derivation ∆ : N × N × N −→ N if F (xy, z, w) = ∆(x, z, w)y + xF (y, z, w)

(resp. F (xy, z, w) = F (x, z, w)y + x∆(y, z, w)) for all w, x, y, z ∈ N . The

map F is called a permuting generalized 3-derivation associated with ∆ if it is

both a permuting left generalized 3-derivation and a permuting right generalized

3-derivation associated with ∆. In this chapter we study generalized n-derivations

in prime near rings.

Section 5.2 contains preliminary results which are required to develop the

proof of our main theorems.

In section 5.3 we investigate some conditions involving traces of generalized

n-derivations in a prime near ring which turn the prime near ring into commuta-

tive ring.


Definition 5.2.1 (Permuting map) Let n ≥ 2 be a fixed positive inte-

76

ger and Nn = N ×N × ....×N︸︷︷︸n−times

. A map ∆ : Nn −→ N is said to be permuting

on a near ring N if relation ∆(x1, x2, ...., xn) = ∆(xπ(1), xπ(2), ...., xπ(n)) holds for

all xi ∈ N , i = 1, 2, ...., n and for every permutation π ∈ Sn.

Definition 5.2.2 (Trace) A map δ : N −→ N defined by δ(x) =

∆(x, x, ...., x) for all x ∈ N , is called trace of ∆, where ∆ : Nn −→ N is a

permuting map.

Very recently Park [162] introduced the notion of n-derivations in ring. Anal-

ogously we define n-derivations in near ring.

Definition 5.2.3 (n-derivation) Let n ≥ 2 be a fixed positive integer.

An n-additive map ∆ : Nn −→ N (additive in each argument) is said to be an

n-derivation on a near ring N if the relations

∆(x1x′1, x2, ...., xn) = ∆(x1, x2, ...., xn)x′1 + x1∆(x′1, x2, ...., xn)

∆(x1, x2x′2, ...., xn) = ∆(x1, x2, ...., xn)x′2 + x2∆(x1, x

′2, ...., xn)

:

∆(x1, x2, ...., xnx′n) = ∆(x1, x2, ...., xn)x′n + xn∆(x1, x2, ...., x

′n)


′2, ....., xn, x

′n ∈ N .

Of course, an 1-derivation is a derivation and a 2-derivation is a bi-derivation.

Remark 5.2.1 Let ∆ be a permuting n-derivation on a near ring N .

(i) ∆(0, x2, ...., xn) = 0 for all x2, x3, ...., xn ∈ N.

77

(ii) ∆(−x1, x2, ...., xn) = −∆(x1, x2, ...., xn) for all x1, x2, x3, ...., xn ∈ N .

Example 5.2.1 Let S be a commutative near ring and let

N ={(

a b0 0

)| a, b, 0 ∈ S

}. Define a map ∆ : Nn −→ N by

∆((

a1 b10 0

),

(a2 b20 0

), ....,

(an bn0 0

))=

(0 a1a2....an0 0

).

It is easy to verify that ∆ is a nonzero permuting n-derivation on the near ring

N .

More recently motivated by the definition of a generalized derivation in near

rings given by Golbasi in [99], Park and Jung [164] defined generalized 3-derivation

in near rings. We define generalized n-derivation in near rings as follows:

Definition 5.2.4 (Generalized n-derivation) Let n ≥ 2 be a fixed

positive integer. An n-additive mapping F : Nn −→ N (additive in each ar-

gument) is said to be a right generalized n-derivation on a near ring N with

associated n-derivation ∆ if the relations

F (x1x′1, x2, ...., xn) = F (x1, x2, ...., xn)x′1 + x1∆(x′1, x2, ...., xn)

F (x1, x2x′2, ...., xn) = F (x1, x2, ...., xn)x′2 + x2∆(x1, x

′2, ...., xn)

:

F (x1, x2, ...., xnx′n) = F (x1, x2, ...., xn)x′n + xn∆(x1, x2, ...., x

′n)


′2, ....., xn, x

′n ∈ N . If in addition both F and ∆ are permut-

ing maps then F is called a permuting right generalized n-derivation of N with

78

associated permuting n-derivation ∆. An n-additive mapping F : Nn −→ N

is said to be a left generalized n-derivation on a near ring N with associated

n-derivation ∆ if the relations

F (x1x′1, x2, ...., xn) = ∆(x1, x2, ...., xn)x′1 + x1F (x′1, x2, ...., xn)

F (x1, x2x′2, ...., xn) = ∆(x1, x2, ...., xn)x′2 + x2F (x1, x

′2, ...., xn)

:

F (x1, x2, ...., xnx′n) = ∆(x1, x2, ...., xn)x′n + xnF (x1, x2, ...., x

′n)


′2, ....., xn, x

′n ∈ N . If in addition both F and ∆ are permut-

ing maps, then F is called a permuting left generalized n-derivation of N with as-

sociated permuting n-derivation ∆. An n-additive mapping F : Nn −→ N is said

to be a generalized n-derivation on a near ring N with associated n-derivation

∆ if it is both a right generalized n-derivation as well as a left generalized n-

derivation of N with associated n-derivation ∆.


N =

{(0 a0 b

)| a, b, 0 ∈ S

}. Obviously N is a near ring under matrix addition

and multiplication. Define maps ∆, F : Nn −→ N by

∆

((0 a10 b1

),

(0 a20 b2

), ....,

(0 an0 bn

))=

(0 a1a2....an0 0

)and

F

((0 a10 b1

),

(0 a20 b2

), ....,

(0 an0 bn

))=

(0 00 b1b2....bn

)respectively.

It can be verified that F is a permuting right generalized n-derivation of N with

associated n-derivation ∆.

79


N =

{(a b0 0

)| a, b, 0 ∈ S

}. Then N is a near ring under matrix addition and

multiplication. Define maps ∆, F : Nn −→ N by

∆

((a1 b10 0

),

(a2 b20 0

), ....,

(an bn0 0

))=

(0 b1b2....bn0 0

)and

F

((a1 b10 0

),

(a2 b20 0

), ....,

(an bn0 0

))=

(a1a2....an 0

0 0

)respectively.

Then F is a left generalized n-derivation of N with associated n-derivation ∆.


N =

{(a 0b c

)| a, b, c, 0 ∈ S

}. N is a near-ring under matrix addition and

multiplication. Define maps ∆, F : Nn −→ N by

∆

((a1 0b1 c1

),

(a2 0b2 c2

), ....,

(an 0bn cn

))=

(0 0

b1b2....bn 0

)and

F

((a1 0b1 c1

),

(a2 0b2 c2

), ....,

(an 0bn cn

))=

(0 0

b1b2....bn 0

)respectively.

Then F is a permuting right generalized n-derivation and a permuting left general-

ized n-derivation of N with associated n-derivation ∆ and hence F is a permuting

generalized n-derivation on N .

Proposition 5.2.1 [163, Lemma 2.4] Let N be a near ring and let ∆ : N3 −→ N

be a permuting 3-derivation. Then N satisfies the following partial distributive

80

law:

[∆(x, z, w)y+x∆(y, z, w)]v = ∆(x, z, w)yv+x∆(y, z, w)v for all v, w, x, y, z ∈ N.

Proof. By hypothesis, we have

∆(xy, z, w) = ∆(x, z, w)y + x∆(y, z, w)

for all w, x, y, z ∈ N and associative law gives

∆((xy)v, z, w) = ∆(xy, z, w)v + xy∆(v, z, w)

= [∆(x, z, w)y + x∆(y, z, w)]v + xy∆(v, z, w) (5.2.1)

for all v, w, x, y, z ∈ N . Also we have

∆(x(yv), z, w) = ∆(x, z, w)yv + x∆(yv, z, w)

= ∆(x, z, w)yv + x[∆(y, z, w)v + y∆(v, z, w)]

= ∆(x, z, w)yv + x∆(y, z, w)v + xy∆(v, z, w) (5.2.2)

for all v, w, x, y, z ∈ N . Comparing (5.2.1) and (5.2.2), we see that

[∆(x, z, w)y + x∆(y, z, w)]v = ∆(x, z, w)yv + x∆(y, z, w)v

for all v, w, x, y, z ∈ N .


ideal of N . If ∆ is a nonzero 3-derivation on N , then ∆ 6= 0 on U .

Proof. Suppose that ∆(U,U, U) = {0}. For any u, v, w ∈ U , we have

∆(u, v, w) = 0. (5.2.3)

81

Substituting ux for u in (5.2.3), we get

∆(u, v, w)x+ u∆(x, v, w) = 0 for all u, v, w ∈ U and x ∈ N.

Using (5.2.3), we get U∆(x, v, w) = {0}. Invoking Lemma 2.2.1(ii), we have

∆(x, v, w) = 0 for all v, w ∈ U and x ∈ N. (5.2.4)

Substituting vy for v in (5.2.4), we get

∆(x, v, w)y + v∆(x, y, w) = 0 for all v, w ∈ U and x, y ∈ N.

Using (5.2.4), we find U∆(x, y, w) = {0} and Lemma 2.2.1(ii), yields that

∆(x, y, w) = 0 for all w ∈ U and x, y ∈ N. (5.2.5)

Substituting wz for w in (5.2.3), we obtain U∆(x, y, z) = {0}. Another appeal to

Lemma 2.2.1(ii), yields that ∆(x, y, z) = 0, for all x, y, z ∈ N , a contradiction.

The following result is an extension of Lemma 2.2 of Park and Jung [164].

Proposition 5.2.3 Let N be a 3!-torsion free near ring and U be a nonzero

additive subgroup of N . If ∆ is a permuting 3-additive map with trace δ such

that δ(x) = 0 for all x ∈ U , then ∆ = 0 on U .

Proof. For any x, y ∈ U , we have the relation

δ(x+ y) = δ(x) + 2∆(x, x, y) + ∆(x, y, y) + ∆(x, x, y) + 2∆(x, y, y) + δ(y)

and so, by the hypothesis, we get

2∆(x, x, y) + ∆(x, y, y) + ∆(x, x, y) + 2∆(x, y, y) = 0 for all x, y ∈ U. (5.2.6)

82

Substituting −x for x in (5.2.6), we obtain

2∆(x, x, y)−∆(x, y, y) + ∆(x, x, y)− 2∆(x, y, y) = 0 for all x, y ∈ U. (5.2.7)

On the other hand, for any x, y ∈ U ,

δ(y + x) = δ(y) + 2∆(y, y, x) + ∆(y, x, x) + ∆(y, y, x) + 2∆(y, x, x) + δ(x)

and thus, by the hypothesis and using the fact that ∆ is permuting, we have

2∆(x, y, y) + ∆(x, x, y) + ∆(x, y, y) + 2∆(x, x, y) = 0 for all x, y ∈ U. (5.2.8)


2∆(x, y, y) + ∆(x, x, y) + ∆(x, y, y) = ∆(x, x, y)− 3∆(x, y, y)

which implies that

2∆(x, y, y)+∆(x, x, y)+∆(x, y, y)+2∆(x, x, y) = ∆(x, x, y)−3∆(x, y, y)+2∆(x, x, y).

Hence it follows from (5.2.8) that

∆(x, x, y)− 3∆(x, y, y) + 2∆(x, x, y) = 0 for all x, y ∈ U. (5.2.9)

Substituting −x for x in (5.2.9), we find

∆(x, x, y) + 3∆(x, y, y) + 2∆(x, x, y) = 0 for all x, y ∈ U. (5.2.10)

Comparing (5.2.9) and (5.2.10), we obtain

6∆(x, y, y) = 0 for all x, y ∈ U.

Since N is 3!-torsion free, we get

∆(x, y, y) = 0 for all x, y ∈ U. (5.2.11)

83

Substituting y + z for y in (5.3.9) and linearizing (5.2.11) we obtain

∆(x, y, z) = 0 for all x, y, z ∈ U,

-i.e., ∆ = 0 on U which completes the proof.


ideal of N . Let ∆ : N3 −→ N be a 3-derivation. If for x ∈ N , ∆(U,U, U)x = {0}

(or x∆(U,U, U) = {0}), then either x = 0 or ∆ = 0 on U .

Proof. Let ∆(y, z, w)x = 0 for all y, z, w ∈ U. Substituting y = vy, we get

∆(v, z, w)yx = 0, for all y, z, v, w ∈ U . Hence ∆(v, z, w)Ux = {0} implies that

either x = 0 or ∆ = 0 by Lemma 2.2.1(i).

In order to extend a result of Mayne [148], Ozturk and Jun [159, Lemma 2]

proved that if D is a permuting bi-derivation on a prime near ring N with trace

d such that xd(y) = 0 for all x, y ∈ N , then either x = 0 or D = 0. Further

Park and Jung [164, Lemma 2.3] obtained the result for a generalized 3-derivation

of N . Now we prove the mentioned result for a generalized 3-derivation in the

setting of a semigroup ideal of N .

Proposition 5.2.5 Let N be a 3!-torsion free 3-prime near ring and U be a

nonzero additive subgroup and a semigroup ideal of N . Let ∆ : N3 −→ N be a

3-derivation.

(i) If F is a permuting right generalized 3-derivation of N associated with ∆ and

x ∈ N such that xf(y) = 0 for all y ∈ U , where f is the trace of F then either

x = 0 or ∆ = 0 on U .

84

(ii) If F is a permuting generalized 3-derivation of N associated with ∆ and

x ∈ N such that xf(y) = 0 for all y ∈ U , where f is the trace of F , then either

x = 0 or F = 0 on U .

Proof. (i) For any y, z ∈ U , we have

f(y + z) = f(y) + 2F (y, y, z) + F (y, z, z) + F (y, y, z) + 2F (y, z, z) + f(z).

Using hypothesis, we obtain

2xF (y, y, z)+xF (y, z, z)+xF (y, y, z)+2xF (y, z, z) = 0 for all y, z ∈ U. (5.2.12)

Substituting −y for y in (5.2.12), it follows that

2xF (y, y, z)−xF (y, z, z)+xF (y, y, z)−2xF (y, z, z) = 0 for all y, z ∈ U. (5.2.13)

On the other hand, for any y, z ∈ U ,

f(z + y) = f(z) + 2F (z, z, y) + F (z, y, y) + F (z, z, y) + 2F (z, y, y) + f(y)

and so, by the hypothesis, we have

2xF (z, z, y) + xF (z, y, y) + xF (z, z, y) + 2xF (z, y, y) = 0

Since F is permuting, we get

2xF (y, z, z)+xF (y, y, z)+xF (y, z, z)+2xF (y, y, z) = 0 for all y, z ∈ U. (5.2.14)


2xF (y, z, z) + xF (y, y, z) + xF (y, z, z) = xF (y, y, z)− 3xF (y, z, z)

85

which implies that

2xF (y, z, z)+xF (y, y, z)+xF (y, z, z)+2xF (y, y, z) = xF (y, y, z)−3xF (y, z, z)+2xF (y, y, z).

Now, from (5.2.14), we obtain

xF (y, y, z)− 3xF (y, z, z) + 2xF (y, y, z) = 0 for all y, z ∈ U. (5.2.15)

Substituting −y for y in (5.2.15), we have

xF (y, y, z) + 3xF (y, z, z) + 2xF (y, y, z) = 0 for all y, z ∈ U. (5.2.16)


6xF (y, z, z) = 0 for all y, z ∈ U.

Since N is 3!-torsion free, we get

xF (y, z, z) = 0 for all y, z ∈ U. (5.2.17)

Substituting z + w for z in (5.2.17), we find that

xF (w, y, z) = 0 for all w, y, z ∈ U. (5.2.18)

Replacing y by yv in (5.2.18) and using (5.2.18), we get xy∆(v, z, w) = 0 for all

v, w, y, z ∈ U . Hence xU∆(v, z, w) = 0 implies that either x = 0 or ∆(v, z, w) = 0

for all v, w, z ∈ U by Lemma 2.2.1(i).

(ii) Let x be a nonzero element of N . Thus from (i) we obtain ∆ = 0. Replac-

ing y by yv in (5.2.18) and using hypothesis, we get xyF (v, z, w) = 0, for all

v, w, y, z ∈ U . Hence xUF (v, z, w) = {0}, again application of Lemma 2.2.1(i)

yields that F = 0 on U .

86

Proposition 5.2.6 Let N be a near ring and let ∆ : N3 −→ N be a 3-derivation.

(i) If F is a right generalized 3-derivation of N associated with ∆, then

[F (x, z, w)y+x∆(y, z, w)]v = F (x, z, w)yv+x∆(y, z, w)v, for all v, w, x, y, z ∈ N .

(ii) If F is a left generalized 3-derivation of N associated with ∆ , then

[∆(x, z, w)y+xF (y, z, w)]v = ∆(x, z, w)yv+xF (y, z, w)v, for all v, w, x, y, z ∈ N .

Proof. (i) Let F be a right generalized 3-derivation of N associated with ∆.

Then

F ((xy)v, z, w) = F (xy, z, w)v + xy∆(v, z, w)

= [F (x, z, w)y + x∆(y, z, w)]v + xy∆(v, z, w)

for all v, w, x, y, z ∈ N. (5.2.19)

On the other hand

F (x(yv), z, w) = F (x, z, w)yv + x∆(yv, z, w)

= F (x, z, w)yv + x[∆(y, z, w)v + y∆(v, z, w)]

= F (x, z, w)yv + x∆(y, z, w)v + xy∆(v, z, w)

for all v, w, x, y, z ∈ N. (5.2.20)


[F (x, z, w)y+x∆(y, z, w)]v = F (x, z, w)yv+x∆(y, z, w)v for all v, w, x, y, z ∈ N.

87

(ii) Let F be a left generalized 3-derivation of N associated with ∆. Then

F ((xy)v, z, w) = ∆(xy, z, w)v + xyF (v, z, w)

= [∆(x, z, w)y + x∆(y, z, w)]v + xyF (v, z, w)

for all v, w, x, y, z ∈ N. (5.2.21)

On the other hand

F (x(yv), z, w) = ∆(x, z, w)yv + xF (yv, z, w)

= ∆(x, z, w)yv + x[∆(y, z, w)v + yF (v, z, w)]

= ∆(x, z, w)yv + x∆(y, z, w)v + xyF (v, z, w)

for all v, w, x, y, z ∈ N. (5.2.22)


[∆(x, z, w)y+xF (y, z, w)]v = ∆(x, z, w)yv+xF (y, z, w)v for all v, w, x, y, z ∈ N.


ideal of N . If F is a nonzero right (or left) generalized 3-derivation of N associ-

ated with a nonzero 3-derivation ∆, then F 6= 0 on U .

Proof. Let F be a nonzero right generalized 3-derivation of N such that

F (U,U, U) = {0}. Then for all x, y, z ∈ U , we have F (x, y, z) = 0. Re-

placing x by xw, we get F (x, y, z)w + x∆(w, y, z) = 0 for w, x, y, z ∈ U .

Hence U∆(w, y, z) = {0} and Lemma 2.2.1(ii) gives that ∆(w, y, z) = 0, for

all w, y, z ∈ U , a contradiction by Proposition 5.2.2 which completes the proof.

88

5.3 Traces of generalized 3-derivations and com-mutativity of prime near rings

In [164] Park and Jung proved that if N is a 2, 3 torsion free prime near ring

admitting a generalized 3-derivation F such that F (x, y, z) ∈ Z, the centre of N

for all x, y, z ∈ N , then N is a commutative ring. We prove the result in case of

a semigroup ideal of N .

Theorem 5.3.1 Let N be a 3!-torsion free 3-prime near ring and U be a nonzero

additive subgroup and a semigroup ideal of N . Suppose ∆ is a permuting 3-

derivation and F is a nonzero permuting generalized 3-derivation of N associated

with ∆ such that f(U) ⊆ U , where f is the trace of F . If F (U,U, U) ⊆ Z, then

N is a commutative ring.

Proof. Suppose that ∆ = 0 on U . Then F (xy, z, w) = F (x, z, w)y ∈ Z for

all w, x, y, z ∈ U gives that F (x, z, w)yv = vF (x, z, w)y for all w, x, y, z ∈ U and

v ∈ N this implies that [y, v]f(x) = 0 for all v, x, y ∈ U . Applying Proposition

5.2.5(ii), we get [y, v] = 0 for all y ∈ U and v ∈ N and U ⊆ Z. Hence N is a

commutative ring by Lemma 2.2.3.

Now, let us consider the case ∆ 6= 0 on U . Using the hypothesis, we get

F (x, y, z)w = wF (x, y, z) for all x, y, z ∈ U and w ∈ N . Replacing x by xv, we

obtain

F (x, y, z)vw+x∆(v, y, z)w = wF (x, y, z)v+wx∆(v, y, z) for all v, x, y, z ∈ U and w ∈ N.

(5.3.1)

Replacing x by rf(x) in (5.3.1) and using (5.3.1), we have

r∆(f(x), y, z)[v, w] = 0 for all r, v, x, y, z ∈ U and w ∈ N. (5.3.2)

89

Thus U∆(f(x), y, z)[v, w] = 0 and by Lemma 2.2.1(ii) ∆(f(x), y, z)[v, w] = 0.

Substituting ry for y, we obtain ∆(f(x), r, z)U [v, w] = {0}. Using Lemma

2.2.1(i), we get either ∆(f(x), y, z) = 0 or [v, w] = 0. Later yields that N is

a commutative ring by Lemma 2.2.3.

Now suppose that

∆(f(x), y, z) = 0 for all x, y, z ∈ U. (5.3.3)

Substituting x+ v for x in (5.3.3) and using the fact that N is 3!-torsion free, we

get that

∆(F (x, v, v), y, z) = 0 for all v, x, y, z ∈ U. (5.3.4)

Replacing v by v + r in (5.3.4), we get

∆(F (x, v, r), y, z) = 0 for all r, v, x, y, z ∈ U. (5.3.5)

Taking xt instead of x in (5.3.5) and using (5.3.5), we have

F (x, v, r)∆(t, y, z)+∆(x, y, z)∆(t, v, r)+x∆(∆(t, v, r), y, z) = 0 for all r, t, v, x, y, z ∈ U.

(5.3.6)

Substituting f(r) for r in (5.3.6) and using (5.3.3),

we get F (f(r), x, v)∆(t, y, z) = 0 for all r, v, t, x, y, z ∈ U. Substituting uv for

v, we obtain F (f(r), x, u)U∆(t, y, z) = {0}. Again by Lemma 2.2.1(i), we get

either F (f(r), x, u) = 0 or ∆(t, y, z) = 0. If ∆(t, y, z) = 0, for all t, y, z ∈ U , then

∆ = 0 on U , a contradiction. Thus we find that

F (f(r), x, v) = 0 for all r, v, x ∈ U. (5.3.7)

Taking r + s instead of r in (5.3.7) and using (5.3.7) and the fact that N is

3!-torsion free, we obtain

F (F (r, r, s), x, v) = 0 for all r, s, x, v ∈ U. (5.3.8)

90

Substituting r + z for r in (5.3.8) and using (5.3.8), we have

F (F (r, z, s), x, v) = 0 for all r, s, v, x, z ∈ U. (5.3.9)

Replacing r by rt in (5.3.9) and using (5.3.9), we get

F (r, z, s)∆(t, z, v)+F (r, x, v)∆(t, z, s)+r∆(∆(t, z, s), x, v) = 0 for all r, s, v, t, x, z ∈ U.

(5.3.10)

Substituting f(r) for r in (5.3.10) and using (5.3.7), we get

f(r)∆(∆(t, z, s), x, v) = 0 for all r, s, v, t, x, z ∈ U − i.e.,

∆(∆(t, z, s), x, v)f(r) = 0 for all r, s, v, t, x, z ∈ U.

Applying Proposition 5.2.5(ii), we obtain

∆(∆(t, z, s), x, v) = 0 for all s, v, t, x, z ∈ U. (5.3.11)

Taking ty instead of t in (5.3.11) and using (5.3.11), we get

∆(t, z, s)∆(y, x, v) + ∆(t, x, v)∆(y, z, s) = 0 for all s, v, t, x, y, z ∈ U. (5.3.12)

Replacing x, z, s, v by t in (5.3.12), we get

δ(t)∆(y, t, t) = 0 for all t, y ∈ U. (5.3.13)

Taking yr instead of y in (5.3.13) and using (5.3.13), we get

91

δ(t)y∆(r, t, t) = 0

and so

δ(t)Uδ(t) = {0} for all t ∈ U.

By Lemma 2.2.1(i), we get δ(t) = 0, for all t ∈ U . Applying Proposition 5.2.3,

we get ∆ = 0 on U , a contradiction.

Ozturk and Yazarli [161, Theorem 3] proved that if N is a 2-torsion free 3-

prime near ring and D1, D2 are nonzero symmetric bi-derivations of N with traces

d1 and d2 respectively such that d2(y), d2(y)+d2(y) ∈ C(D1(x, z)), the centralizer

of D1(x, z) for all x, y, z ∈ N , then (N,+) is abelian and d2(N) ⊆ Z. Later in

[160, Theorem 5] they obtained that if N is a 2, 3-torsion free prime near ring

and F is a nonzero permuting right generalized 3-derivation of N with associated

3-derivation D such that f(x), f(x) + f(x) ∈ C(D(y, z, w)) for all w, x, y, z ∈ N ,

where f is the trace of F , then (N,+) is abelian and f(N) ⊆ Z. Motivated by

the aforementioned results we prove the following:


additive subgroup and a semigroup ideal of N . Suppose ∆ is a nonzero 3-

derivation of N and F is a nonzero permuting right generalized 3-derivation of

N associated with ∆. If f(x), f(x) + f(x) ∈ C(∆(y, z, w)), for all w, x, y, z ∈ U ,

where f is the trace of F , then (N,+) is abelian. Moreover if δ(U) ⊆ U , where δ

is the trace of ∆, then f(U) ⊆ Z.

Proof. For all v, w, x, y, z ∈ U

92

∆(v + y, z, w)(f(x) + f(x))

= (f(x) + f(x))∆(v + y, z, w)

= (f(x) + f(x))[∆(v, z, w) + ∆(y, z, w)]

= (f(x) + f(x))∆(v, z, w) + (f(x) + f(x))∆(y, z, w)

= ∆(v, z, w)(f(x) + f(x)) + ∆(y, z, w)(f(x) + f(x))

= ∆(v, z, w)f(x) + ∆(v, z, w)f(x) + ∆(y, z, w)f(x)

+ ∆(y, z, w)f(x)

= f(x)∆(v, z, w) + f(x)∆(v, z, w) + f(x)∆(y, z, w)

+ f(x)∆(y, z, w)

for all v, w, x, y, z ∈ U. (5.3.14)

On the other hand

∆(v + y, z, w)(f(x) + f(x))

= ∆(v + y, z, w)f(x) + ∆(v + y, z, w)f(x)

= f(x)∆(v + y, z, w) + f(x)∆(v + y, z, w)

= f(x)[∆(v, z, w) + ∆(y, z, w)]

+ f(x)[∆(v, z, w) + ∆(y, z, w)]

= f(x)∆(v, z, w) + f(x)∆(y, z, w) + f(x)∆(v, z, w)

+ f(x)∆(y, z, w)

for all v, w, x, y, z ∈ U. (5.3.15)


f(x)∆((v, y), z, w) = 0 for all v, w, x, y, z ∈ U.

93

By hypothesis we get

∆((v, y), z, w)f(x) = 0 for all v, w, x, y, z ∈ U.

Hence it follows from Proposition 5.2.5(i), that

∆((v, y), z, w) = 0 for all v, w, x, y, z ∈ U.

Replacing v by sv and y by sy, we get ∆(s, z, w)(v, y) = 0, for all s ∈ U . Re-

placing v by vr and y by vp, we get ∆(s, z, w)v(r, p) = 0, for all s, v, w, z ∈ U

and r, p ∈ N -i.e., ∆(s, z, w)U(r, p) = {0}. Using Lemma 2.2.1(i), we get either

∆(s, z, w) = 0 or (r, p) = 0. If ∆(s, z, w) = 0, for all s, z, w ∈ U , then ∆ = 0 on

U , a contradiction by Proposition 5.2.1. Hence (r, p) = 0, for r, p ∈ N and (N,+)

is abelian.

Since f(x) ∈ C(∆(y, z, w)), for all x, y, z, w ∈ U , we have

f(x)∆(y, z, w) = ∆(y, z, w)f(x) for all x, y, z, w ∈ U. (5.3.16)

Replacing y by yv in (5.3.16), we obtain

f(x)∆(y, z, w)v + f(x)y∆(v, z, w) = ∆(y, z, w)vf(x) + y∆(v, z, w)f(x) (5.3.17)

Replacing y by δ(y) in (5.3.17), and using the hypothesis, we get

∆(δ(y), z, w)[v, f(x)] = 0 for all v, w, x, y, z ∈ U. (5.3.18)

Substituting zt for z in (5.3.18), we get

∆(δ(y), z, w)t[v, f(x)] = 0 for all t, v, w, x, y, z ∈ U

-i.e.,

∆(δ(y), z, w)U [v, f(x)] = 0 for all v, w, x, y, z ∈ U.

94

By Lemma 2.2.1(i), we get either ∆(δ(y), z, w) = 0 or [f(x), v] = 0, for

v, w, x, y, z ∈ U . Suppose that ∆(δ(y), z, w) = 0 for all y, z, w ∈ U . Taking

y + v instead of y, we get

∆(δ(y), z, w) + ∆(δ(v), z, w) + 3∆(∆(y, y, v), z, w) + 3∆(∆(y, v, v), z, w) = 0.

Since ∆(δ(y), z, w) = 0 and N is 3!-torsion free, we get

∆(∆(y, y, v), z, w) + ∆(∆(y, v, v), z, w) = 0 for all v, w, y, z ∈ U. (5.3.19)

Replacing y by −y in (5.3.19), we have

∆(∆(y, y, v), z, w)−∆(∆(y, v, v), z, w) = 0 for all v, w, y, z ∈ U. (5.3.20)


∆(∆(y, v, v), z, w) = 0 for all v, w, y, z ∈ U. (5.3.21)

Replacing y by yx in (5.3.21) and using (5.3.21), we have

∆(y, v, v)∆(x, z, w) + ∆(y, z, w)∆(x, v, v) = 0 for all v, w, x, y, z ∈ U.

Taking xt instead of x in the above relation, we get

∆(y, v, v)x∆(t, z, w) + ∆(y, z, w)x∆(t, v, v) = 0 for all t, v, w, y, z ∈ U. (5.3.22)

Replacing t, w, y, z by v in (5.3.22), we get δ(v)xδ(v) = 0, for all x, v ∈ U -

i.e., δ(v)Uδ(v) = {0}. By Lemma 2.2.1(i), we get δ(v) = 0, for all v ∈ U .

Proposition 5.2.3 yields that ∆ = 0 on U , a contradiction by Proposition 5.2.1.

Thus [f(x), v] = 0, for all v, x ∈ U . Replacing v by vr, for all r ∈ N , we get

v[f(x), r] = 0 -i.e., U [f(x), r] = 0 for all x ∈ U and r ∈ N . Again by Lemma

2.2.1(ii), we get [f(x), r] = 0 for all x ∈ U and r ∈ N . Hence f(U) ⊆ Z.

95


additive subgroup and a semigroup ideal of N . Let ∆ be a 3-derivation on N . If

F is a nonzero permuting generalized 3-derivation of N associated with ∆ such

that f(x), f(x) + f(x) ∈ C(F (u, v, w)), for all u, v, w, x ∈ U , where f is the trace

of F , then (N,+) is abelian.

Proof. For all p, u, v, w, x ∈ U

F (u+ p, v, w)(f(x) + f(x))

= (f(x) + f(x))F (u+ p, v, w)

= (f(x) + f(x))[F (u, v, w) + F (p, v, w)]

= (f(x) + f(x))F (u, v, w) + (f(x) + f(x))F (p, v, w)

= F (u, v, w)(f(x) + f(x)) + F (p, v, w)(f(x) + f(x))

= F (u, v, w)f(x) + F (u, v, w)f(x) + F (p, v, w)f(x)

+ F (p, v, w)f(x)

= f(x)F (u, v, w) + f(x)F (u, v, w) + f(x)F (p, v, w)

+ f(x)F (p, v, w)

for all p, u, v, w, x ∈ U. (5.3.23)

On the other hand

F (u+ p, v, w)(f(x) + f(x)) = F (u+ p, v, w)f(x) + F (u+ p, v, w)f(x)

= F (u+ p, v, w)f(x) + F (u+ p, v, w)f(x)

= f(x)F (u+ p, v, w) + f(x)F (u+ p, v, w)

96

= f(x)[F (u, v, w) + F (p, v, w)]

+ f(x)[F (u, v, w) + F (p, v, w)]

= f(x)F (u, v, w) + f(x)F (p, v, w) + f(x)F (u, v, w)

+ f(x)F (p, v, w)

for all p, u, v, w, x ∈ U. (5.3.24)


f(x)F ((u, p), v, w) = 0 for all p, u, v, w, x ∈ U.

By hypothesis we get

F ((u, p), v, w)f(x) = 0 for all p, u, v, w, x ∈ U.

Hence it follows from Proposition 5.2.6(ii), that

F ((u, p), v, w) = 0 for all p, u, v, w ∈ U. (5.3.25)

First, let us consider the case ∆ = 0. Substituting uz for u and up for p in

(5.3.25), we get

F (u, v, w)(z, p) + u∆((z, p), v, w) = 0 for all p, u, v, w, z ∈ U. (5.3.26)

Thus

F (u, v, w)(z, p) = 0 for all p, u, v, w, z ∈ U. (5.3.27)

Replacing z by zr and p by zs in (5.3.27), we get

F (u, v, w)z(r, s) = 0 for all u, v, w, z ∈ U and r, s ∈ N

-i.e.,

F (u, v, w)U(r, s) = 0 for all u, v, w ∈ U and r, s ∈ N. (5.3.28)

97

Invoking Lemma 2.2.1(i), either (r, s) = 0 or F (u, v, w) = 0. Later yields a con-

tradiction by Proposition 5.2.7. Hence (r, s) = 0 for all r, s ∈ N and (N,+) is

abelian.

Now, let us consider the case ∆ 6= 0. Again substituting uz for u and up for

p in (5.3.25), we get

∆(u, v, w)(z, p) + uF ((z, p), v, w) = 0 for all p, u, v, w, z ∈ U.

Using (5.3.25) we have

∆(u, v, w)(z, p) = 0 for all p, u, v, w, z ∈ U. (5.3.29)

Replacing z by zr and p by zs in (5.3.29), we obtain

∆(u, v, w)z(r, s) = 0 for all u, v, w, z ∈ U and r, s ∈ N

-i.e.,

∆(u, v, w)U(r, s) = {0} for all u, v, w ∈ U and r, s ∈ N. (5.3.30)

Applying Lemma 2.2.1(i), we get either (r, s) = 0 or ∆(u, v, w) = 0. Later yields

a contradiction by Proposition 5.2.2. Hence (r, s) = 0 for all r, s ∈ N and (N,+)

is abelian.


additive subgroup and a semigroup ideal of N . Suppose ∆ is a 3-derivation on

N and F is a nonzero permuting generalized 3-derivation of N associated with ∆

such that f(U) ⊆ U and δ(U) ⊆ U , where f and δ are the trace of F and trace

98

of ∆ respectively. If f(x), f(x)+f(x) ∈ C(F (u, v, w)), for all u, v, w, x ∈ U , then

N is a commutative ring.

Proof. First, let us consider the case ∆ = 0. Then

F (xy, z, t) = F (x, z, t)y + x∆(y, z, t)

and we have

F (xy, z, t) = F (x, z, t)y for all t, x, y, z ∈ U. (5.3.31)

On the other hand

F (xy, z, t) = ∆(x, z, t)y + xF (y, z, t),

and we have

F (xy, z, t) = xF (y, z, t) for all t, x, y, z ∈ U. (5.3.32)

Comparing (5.3.31) and (5.3.32) we obtain

F (x, z, t)y = xF (y, z, t) for all t, x, y, z ∈ U. (5.3.33)

Replacing x by y in (5.3.33), we obtain

[F (y, z, t), y] = 0 for all t, y, z ∈ U. (5.3.34)

Substituting zr for z in (5.3.34), we get

[F (y, z, t)r, y] = 0 for all t, y, z ∈ U and r ∈ N.


F (y, z, t)[r, y] = 0 for all t, y, z ∈ U and r ∈ N. (5.3.35)

Substituting rs for r in (5.3.35) and using (5.3.35), we get

F (y, z, t)r[s, y] = 0 for all t, y, z ∈ U and r, s ∈ N

99

-i.e.,

F (y, z, t)N [s, y] = 0 for all t, y, z ∈ U and s ∈ N.

Since N is a 3-prime near ring, then either [s, y] = 0 or F (y, z, t) = 0. Later

yields a contradiction by Proposition 5.2.7. Thus [s, y] = 0 for all y ∈ U and

s ∈ N . Hence U ⊆ Z and N is a commutative ring by Lemma 2.2.3.

Now, let ∆ 6= 0. By hypothesis

[f(x), F (u, v, w)] = 0 for all u, v, w, x ∈ U. (5.3.36)

Replacing x by x+ y in (5.3.36) and applying Theorem 5.3.3, we obtain

[F (x, x, y), F (u, v, w)] + [F (x, y, y), F (u, v, w)] = 0 for all u, v, w, x, y ∈ U.

(5.3.37)

Setting y = −y in (5.3.37) and comparing the result with (5.3.37), we obtain

[F (x, y, y), F (u, v, w)] = 0 for all u, v, w, x, y ∈ U. (5.3.38)

Replacing y by y + z in (5.3.38) and using (5.3.38) and the fact that F is per-

muting, we have

[F (x, y, z), F (u, v, w)] = 0 for all u, v, w, x, y, z ∈ U

-i.e.,

F (x, y, z)F (u, v, w) = F (u, v, w)F (x, y, z) for all u, v, w, x, y, z ∈ U. (5.3.39)

Substituting ut for u in (5.3.39) and applying Proposition 5.2.4, we obtain

∆(u, v, w)tF (x, y, z)− F (x, y, z)∆(u, v, w)t+ uF (t, v, w)F (x, y, z)

−F (x, y, z)uF (t, v, w) = 0 for all t, u, v, w, x, y, z ∈ U. (5.3.40)

100

Substituting f(u) for u in (5.3.40) and then using hypothesis and (5.3.40), we get

∆(f(u), v, w)tF (x, y, z)− F (x, y, z)∆(f(u), v, w)t = 0 (5.3.41)

Now replacing t by f(t), we have

∆(f(u), v, w)f(t)F (x, y, z)− F (x, y, z)∆(f(u), v, w)f(t) = 0− i.e.,

∆(f(u), v, w)F (x, y, z)f(t)− F (x, y, z)∆(f(u), v, w)f(t) = 0.

Applying Proposition 5.2.1, we get

[∆(f(u), v, w)F (x, y, z)− F (x, y, z)∆(f(u), v, w)]f(t) = 0

Using Proposition 5.2.5(ii)

∆(f(u), v, w)F (x, y, z) = F (x, y, z)∆(f(u), v, w) for all u, v, w, x, y, z ∈ U

and (5.3.41) yields that

∆(f(u), v, w)[t, F (x, y, z)] = 0 for all t, u, v, w, x, y, z ∈ U. (5.3.42)

Replacing w by ww′ in (5.3.42), we have

∆(f(u), v, w)w′[t, F (x, y, z)] = 0 for all t, u, v, w, w′, x, y, z ∈ U

-i.e.,

∆(f(u), v, w)U [t, F (x, y, z)] = 0 for all t, u, v, w, x, y, z ∈ U.

From Lemma 2.2.1(i), either ∆(f(u), v, w) = 0 or [t, F (x, y, z)] = 0. If

∆(f(u), v, w) = 0 for all u, v, w ∈ U, then arguing in the similar manner as

in the proof of Theorem 5.3.1, we arrive at a contradiction. Thus

[t, F (x, y, z)] = 0 for all t, x, y, z ∈ U. (5.3.43)

101

Substituting tr′ for t in (5.3.43), we get

t[r′, F (x, y, z)] = 0 for all t, x, y, z ∈ U and r′ ∈ N

-i.e.,

U [r′, F (x, y, z)] = 0 for all x, y, z ∈ U and r′ ∈ N. (5.3.44)

By Lemma 2.2.1(ii), we get [r′, F (x, y, z)] = 0 for all x, y, z ∈ U and r′ ∈ N and

F (x, y, z) ∈ Z for all x, y, z ∈ U and N is a commutative ring by Theorem 5.3.1.

The following examples show that the conditions in the hypothesis of

Theorem 5.3.2 – Theorem 5.3.4 are not superfluous.

Example 5.3.1 In example 5.2.2 if we take U to be the set of all matrices of

the type

(0 00 b

), then U is an additive subgroup but not a semigroup ideal

of N . N is not a 3-prime near ring. It can be easily verified that δ(U) ⊆ U ,

f(x), f(x)+f(x) ∈ C(∆(u, v, w)), for all u, v, w, x ∈ U . But (N,+) is not abelian

and f(U) 6⊆ Z. Hence N to be 3-prime near ring and U to be a semigroup ideal

are essential in the hypothesis of Theorem 5.3.2.

Example 5.3.2 In Example 5.2.4 if we take U to be the set of all matri-

ces of the type

(a 00 0

). Then U is an additive subgroup but not a semi-

group ideal of N . N is not a 3-prime near ring and δ(U) ⊆ U , f(U) ⊆ U ,

f(x), f(x) + f(x) ∈ C(F (u, v, w)), for all u, v, w, x ∈ U and F (U,U, U) ⊆ Z. But

neither (N,+) is abelian nor N under multiplication is commutative. Hence N to

be 3-prime near ring and U to be a semigroup ideal are essential in the hypothesis

of Theorem 5.3.3 and Theorem 5.3.4.

102

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