ANIJMIT VOL 7-11 2010-2014

Acharya Nagarjuna International Journal of Mathematics

and Information Technology

Vol : 7 2010 Vol : 8 2011 Vol : 9 2012 Vol : 10 2013 Vol : 11 2014

ISSN 0973 - 3477

Acharya Nagarjuna University

Nagarjuna Nagar - 522 510 Andhra Pradesh, India.

Devoted to Original Research in Pure &

Applied Mathematics, Statistics and related

Mathematical Sciences & IT

ISSN 0973-3477_____________________________________________

Acharya Nagarjuna InternationalJournal of Mathematics

and Information Technologye-mail: [email protected]

Vol: 7 ∙ 2010 , Vol: 8 ∙ 2011, Vol: 9 ∙ 2012, Vol: 10 ∙ 2013, Vol: 11 . 2014

CONTENTSResearch Papers Page No

Volume ∙7

Viscous Dissipation of Energy due to slow motion of Porous 01-13 Newtonian Sphere in Visco-Elastic Fluid.

Jitesh Kumar Singh & Nand Lal Singh

Volume ∙8A Note on Weighted Ostrowski – Gruss type inequality 01 - 10 and Applications.

A. Qayyum

Volume ∙09g -Normal Spaces Almost g -Normal And 01-20

Mildly g -Normal Spaces

C. Janaki & Sr. I. Arockiarani

Volume ∙10

. The Behavior of Some Modules in ][M 01-12 Over an HNP Ring

IrawatiVolume ∙11. Generalized Closed Sets with Respect to an Ideal 01-07 R. Algar

Acharya Nagarjuna International Journal of Mathematics & Information TechnologyISSN 0973-3477Vol. 7 PP 01-13

ANIJMITAcharya Nagarjuna

University – 2010

VISCOUS DISSIPATION OF ENERGY DUE TO SLOW MOTION OF POROUS NEWTONIAN

SPHERE IN VISCO-ELASTIC FLUID

Jitesh Kumar SinghDepartment of Mathematics, T. D. Postgraduate College,

Jaunpur – 222002, INDIA.

and

Nand Lal SinghDepartment of Mathematics, T. D. Postgraduate College,

Jaunpur – 222002, INDIA.

(Communicated by Prof. Dr. S. Sreenadh, SVU)

Abstract

In the present problem we have studied the effect of porosity and particle concentration on viscous dissipation of mechanical energy. Newtonian porous inclusions, spherical in shape, are dilutely suspended in Rivlin – Ericksen visco-elastic fluid. Limiting cases, for rigid spheres and gaseous bubbles, are analysed.

Key words: Rivlin–Ericksen visco-elastic fluid; Newtonian porous

sphere; dissipative mechanical energy.

AMS Classification: 76 A 05, 80 A 20.

1. Introduction

It is of general realization that whenever a solid particle moves in

an ambient fluid, it experiences drag on its surface. The particle

translates and rotates with the surrounding fluid so that only the pure

shearing motion gives rise to a disturbance flow. The total rate of

2 Jitesh Kumar Singh, Nand Lal Singh

dissipation is increased and hence the effective viscosity of suspension is

greater than the viscosity of the ambient fluid. Einstein empirical

formula [3, 4] relates the viscosity of suspension to the viscosity of

suspending fluid for small volume concentration of the rigid particles.

The problem becomes more realistic (as per physiological flow in

capillaries) when the Newtonian content bounded by the porous

membrane is suspended in non-Newtonian fluid. Shmakov and

Shmakova [11, 12], Singh [13, 14] have obtained the effective viscosity

of suspension when the spherical particles are mixed in Power-law,

Reiner – Rivlin and Rivlin – Ericksen fluids, respectively. Sun and

Jayaraman [15] derived the theoretical relation for the bulk stress in

dilute suspension of neutrally buoyant, uniform size, spherical drops in a

visco-elastic liquid medium. The disturbance velocity and pressure fields

interior and exterior to second order fluid drop suspended in a simple

shear flow of another second order fluid were derived by Peery [8] for

small Wiessenberg number, omitting inertia terms. Ramkissoon [9]

calculated the drag force experienced by spherical particle moving in

micropolar fluid medium. Non-Newtonian fluid flow past porous

permeable boundary have been studied by Shafie et al. [10], Cortell [1],

Degan et al. [2] and Ishak et al. [5].

In the present paper, we have obtained the theoretical expression

for the dissipative energy of dilute suspension of neutrally buoyant,

uniform size, porous spherical particles. The particles are of such small

linear dimension that (i) the effect of gravity and inertia on the motion of

the particle are negligible so that a particle moves with the ambient fluid

locally, (ii) the Reynolds number of the disturbed motion resulting from

the presence of the particle is small compared with unity. Results are

discussed with the help of tables.

Viscous Dissipation Of Energy … 3

2. Mathematical Analysis

The flow of visco-elastic Rivlin – Ericksen fluid past a porous

spherical particle filled with Newtonian fluid is considered. The origin of

co-ordinate lies on the centre of the particle. The continuity and

momentum equations for the steady viscous flow neglecting inertia terms

are

, 0i jv (2.1)

, 0i j j (2.2)

where 1 3 2i j i j i j i k k j i jp e e e b ,

, ,i j i j j ie v v ,

2i j i j j i i m m jb a a v v ,

,i

i j i j

va v v

t

,

where i, j, k = r, , : x, y, z; i j = stress tensor, v = velocity vector,

ai = acceleration vector, 1 = Newtonian viscosity, 2 = visco-elastic

coefficient, 3 = cross-viscosity coefficient, i j = Kronecker delta

function and p = pressure.

Neglecting inertia terms, the equation (2.2) in curvilinear co-

ordinate takes the form

1 2 3 2 3 1 3 1 21 2 3 1 2 3

10 F h h h h h h

h h h q q q

(2.3)

In spherical polar co-ordinate system


1 2 3 1 2 31, , sin ; , ,h h r h r q r q q . (2.4)

Under zero body force (F = 0), equation (2.3), in view of (2.4) for

axisymmetric motion reduces to

1 10 2 cotr r r

r r rr r r

(2.5)

1 10 cot 3r

rr r r

. (2.6)

Continuity equation (2.1) reduces to

2 1cot 0r

r

v vvv

r r r r

(2.7)

Let2

1

sinrvr

,

1

sinv

r r

(2.8)

where (vr, v, 0) is the velocity at the point (r, , ) and is the stream

function. We introduce the following non-dimensional quantities

21

323

3 1

, ,

, ,

, ,

ii

i j i j i ji j i j i j

vr pv p

a aq p

e be b

q q q

qa q

(2.9)

q is a constant of dimension T–1.

3. Solution

For dilute suspension we may assume:


(1) 2 (2)

(0) (1) 2 (2)

(0) ... ... ...

... ... ...

p p p p

(3.1)

.

Under zero order approximation equations (2.5) and (2.6) reduce to :

22

(0)2 2

sin 10

sin

(3.2)

The solution of equation (3.2) is obtained as

(0) 2( ) sin cosf (3.3)

where 3 52

Bf A C D

. (3.4)

In the case of uniform flow at infinity we must have Lamb [7]

(0) 3 21sin cos

2 (3.5)

Hence, on comparing (3.3) with (3.5) we find

D = 0 and1

2C .

The external Newtonian flow field is given by

(0) 3 22

1sin cos

2e

BA

(3.6)

and the internal flow field within the sphere is given by

(0) 3 5 2sin cosi C D (3.7)

Four constants A, B, C, D of equations (3.6) and (3.7) are determined

from the conditions that radial and transverse velocities as well the

shearing stresses, at inner and outer surfaces of the spherical particle are

equal.


Thus,

( ) ( )

2 2( ) ( )

( ) ( )

2 cos sin2

; , at = 1i e

e i

i e

mv v

(3.8)

where,( )

( )

2

1 1

sini

ii

and( )

( )1 2

1 1

sine

e

give 2 5

14 1

A m

,

3 1

4 1

mB

,

3 5 2

4 1

mC

,

3 1,

4 1

mD

1

i

(3.9)

where, i = viscosity of particle’s material and m = porosity parameter.

For first order approximation in , we have

22

(1)2 2

sin 1

sin e

22

4 7

576 1800 AB

9

1920 1AB

2sin cos

22

4 7

720 11400 AB

9

2160 1AB

4sin cos

(3.10)

and

22

(1)( )2 2

sin 10

sin i

(3.11)


Put (1) 2 4( ) 1 2( ) sin cos ( ) sin cose f f (3.12)

(1) 2 4( ) 3 4( ) sin cos ( ) sin cosi f f (3.13)

Provided

22

1 1 12 3 4 7

576 112 24 800iv ABf f f

2 2 29 2 3 4

1920 1 16 32 160A Bf f f

22

2 2 2 22 3 4 4 7

720 140 80 280 1400iv ABf f f f

9

2160 1A B

and 3 3 3 4 4 42 3 2 3 4

12 24 16 32 160ivf f f f f f

4 4 4 42 3 4

40 80 2800ivf f f f

.

In view of these equations and (1) 2O when 1, the

solutions of equations (3.10) and (3.11) in the light of equations (3.12)

and (3.13) are

2(1) 24( ) 3 2 3 5

4 1 4 1sin cose

A ABAA

223 4

2 4 3 5

9 1 6 15

A ABB BB

4sin cos

(3.14)

(1) 5 3 2( ) 1 2 sin cosi A A 7 5 4

1 2 sin cosB B

(3.15)


To determine the eight arbitrary constants A3, A4, B3, B4, A1, A2, B1, B2

we assume the kinematical condition at the interfacial boundary as :

( ) ( )i ev v ,

2 2 2 4( ) ( ) 2 cos sin 4 sin 5 sin

2e i

mv v ,

(1) (1)( ) ( ) (0) (0)

2 2

1 1e e e e

sin at = 1.

(3.16)

The super scripts (0) and (1) denote the quantities for zero and

first order smallness and the subscripts (e) and (i) denote external and

internal flows, respectively. Further the analysis is simplified by

considering those terms of the external flow field which lead to give the

rheological behaviour up to first order volume concentration of particles

in the suspending medium. Velocity and pressure are obtained as :

(0) 22

2 3 sin2

Av

(3.17)

(0) 3sin cos

2v

(3.18)

(0) 3sin cos

2v

(3.19)

(1) 2 2 2 432

12 3 sin 5 4 sin 5 sinv A B

(3.20)

(1) 0v

(1) 233

1 804 5 13 4

3p A A B


2 236 17 14 6 140 sinA A B

2 2

440030 3 7 sin

3

BA

(3.21)

23

1 2510 1 30 1

5 1 4A A AB m

216 1 60 1A AB 2 515 20

2

mB B

.

We compute the rate of dissipation of mechanical energy per unit

volume due to external forces acting on surface S as :

S

Wv v dS

V (3.22)

(0) (0) (0) (0) (0) (1)3

S

Wv v v

V

( ) (0) (0) (1) (1) (0)i v v v dS (3.23)

where,2 2(1) (1) (0) (0) (0)p e e e b ,

(0)(0) (0) (0) (0) 2

vp e p

,

(1) (1) (0) (0) (0)e e e b .

After a lengthy tedious calculation we obtain

21

23 1 1

5

Wq A

V

23

2 32 52

5 63 7A A B A

(3.24)


where 34

3V R ,

3

3

4343

a

R

, *

WW

V

where a is the radius of the spherical particles and R (R >> r) is the

radius of bulk fluid sphere.

4. Discussion

For = 0, = 0, m = 0 we have A3 = – 35/4. So the equation (3.24) for

purely Newtonian fluid reduces to W* = 6 1 (qa)2 a (4.1)

which is a classical results for the spheres of radius ‘a’ moving in an

Newtonian infinite fluid of stream velocity ‘qa’.

Occurrence of second term in the right hand side of equation

(3.24) gives the effect of visco-elastic medium on W*. We consider the

two particular cases, one for solid sphere and the other for gas bubble.

Case – I: For solid spheres : ,

2

3

5 31 , 1

4 435 5

1 14 4

A m B m

A m m

(4.2)

and equation (3.24) becomes

21

1* 3 1

2

mW q

132 791

14

m

333 1 7 4q m . (4.3)

From equation (4.3) we observe that for 0.6 < m < 1.0 the value of W*

increases with .


Case – II : For gas bubbles : 0,

11

2A m , B = 0

23

41 1

5 2

mA m

and equation (3.24) becomes

21

1* 3 1

5

mW q

32

242 1863 1

175

mq

3

3

16 133 1

25

mq

(4.4)

Equation (4.4) shows that W* increases with for 0 < m < 1. Hence the

analysis leads to the conclusion that the rate of viscous dissipation of

mechanical energy due to visco-elastic behaviour of the medium

increases with for permeable porous boundary surface. Kawase and

Ulbrecht [6] also predicated that the non-Newtonian effect is more

pronounced at large porosity parameter for power law fluid. Our

equation (3.24) is a general formula sufficient enough to derive the result

for Newtonian and Reiner – Rivlin fluids by taking 3 = 0, = 0 and

= 0, respectively. Variations of porous solid and gaseous spherical

inclusions with respect to and m are given in Tables – 1 and 2.

For Solid SpheresTable – 1

W*

m = 0 m = 0.2 m = 0.40.1 1.1187 1.1266 1.12740.2 1.1174 1.1333 1.13480.3 1.1161 1.1399 1.14230.4 1.1148 1.1466 1.14970.5 1.1135 1.1533 1.1572


For Gas BubblesTable – 2

W*

m = 0 m = 0.2 m = 0.40.1 1.1326 1.1291 1.12580.2 1.1453 1.1382 1.13170.3 1.1580 1.1473 1.13760.4 1.1707 1.1565 1.14350.5 1.1834 1.1656 1.1494

References

[1]. CORTELL R. "Flow and Heat Transfer of an Electrically Conducting Fluid of Second Grade Over a Stretching Sheet Subject to Suction and to a Transverse Magnetic Field" International Journal of Heat and Mass Transfer, 49 (2006), 1851-1856.

[2]. DEGAN G., AKOWANOU C. & AWANOU N. C. "Transient Natural Convection of Non-Newtonian Fluids about a Vertical Surface Embedded in an Anisotropic Porous Medium"International Journal of Heat and Mass Transfer, 50 (2007) 4629-4639.

[3]. EINSTEIN A."Über die Von der Molekular – Kinetischen Theorie der Ẅarme Geforderte Bewegung Von in Ruhenden Flussigkeiten suspendierten Teilehen" Ann. Phys., Bd 19 (1906).

[4]. EINSTEIN A. & Berichtigung Zu Meiner Arbeit "Eine Neue Bestimmung Der Malekuldimensionen" Ann. Phys., Bd 34 (1911).

[5]. ISHAK A., NAZAR R. & POP I. "Boundary Layer Flow of a Micropolar Fluid on a Continuously Moving or Fixed Permeable Surface" International Journal of Heat and Mass Transfer, 50(2007), 4743 – 4748.

[6]. KAWASE Y. & ULBRECHT J. J. "A Power Law Fluid Flow Past a Porous Sphere" Rheologica Acta, 20 (1981), 128-132.


[7]. LAMB H. "Hydrodynamics", Cambridge University Press, p 603 (1945).

[8]. PEERY J. H. "Ph. D. Thesis", Princeton University (1966).

[9]. RAMKISSOON H. "Flow of Micro-polar Fluid Past a Newtonian Fluid Sphere" ZAMM, 65(16) (1985), 635-637.

[10]. SHAFIE S., AMIN N. & POP I. "Unsteady Boundary Layer due to a Stretching Sheet in a Porous Medium using Brinkman Equation Model" J. Heat and Technology, 25(2) (2006), 111-117.

[11]. SHMOKOV U. I. & SHMOKOVA L. M. "Viscosity of Dilute Suspension of Spherical Particles Suspended in Non-Newtonian Fluid" Journal Prekladnoi Mechenik and Technical Physics, 5(1977), 81-85.

[12]. SHMOKOVA L. M. "Rheological Behaviour of Dilute Suspension of Spherical Particle in Non-Newtonian Fluid" Journal Prekladnoi Mechenik and Technicol Physics, 6 (1978), 84-88.

[13]. SINGH N. L. "Rheology of Dilute Suspension of Spherical Particles Suspended in Visco-elastic Fluid". Ind. Jour. Theo. Physics, 37(2) (1989), 155-164.

[14]. SINGH N. L. "Flow of Visco-elastic Fluid Past a Porous Sphere Filled with Newtonian Fluid". Jour. PAS, 5 (1996), 1-9.

[15]. SUN K. & JAYARAMAN K. "Bulk Rheology of Dilute Suspension in Visco-elastic Liquid" Rheologica Acta, 23 (1984), 84-89.

Acharya Nagarjuna International Journal of Mathematics & Information TechnologyISSN 0973-3477Vol.8 PP 01-10


University – 2011

(Communicated by Prof. Dr. Geetha S. Rao)

2 Qayyum

A Note on Weighted Ostrowski-Gruss type … 3

4 Qayyum

6 Qayyum

8 Qayyum

10 Qayyum



University – 2012

g -NORMAL SPACES ALMOST g -NORMAL AND

MILDLY g -NORMAL SPACES

C. JanakiDepartment of Mathematics, Sree Narayana Guru College,

Coimbatore -105, Tamilnadu, INDIA.

and

Sr. I. ArockiaraniDepartment of Mathematics, Nirmala College for Women,

Coimbatore – 18, Tamilnadu, INDIA.

(Communicated by Prof. Dr. Balachandran, Baratiyar Univ.,)

Abstract

In this paper, we define new separate axioms called g - normal, almost g - normal spaces using g -open sets.

Further characterization of almost -normal and mildly- -normal spaces are obtained.

Key words: M - g -open, g -normal, almost g -normal, midly g - normal g -irresolute.

1. Introduction

Levine [6] initiated the investigation of so called g -closed sets in

topological spaces. Since then many modifications of g -closed sets were

defined and investigated by a large number of topologists. Continuity,

compactness, connectedness and separation axioms on topological

2 Janaki, Arochiarani

spaces is an important and basic subject in studies of General topology.

In 1943, Singal and Singal [16] introduced a weak form of normal spaces

called mildly normal spaces. In 1989, Nour [14] used pre-open sets to

define P -normal spaces. R.Devi [3] used -open sets to define

-normal spaces. Navalgi [11] continued the study of further properties

of P -normal spaces and also defined mildly P -normal spaces.

In this paper, we introduce new separation axioms called

g -normal, almost g -normal, mildly g -normal spaces and

obtain a characterization of almost -normal, mildly -normal spaces

using g -open sets.

2. Preliminaries

Throughout this paper, ,X and ,Y (or simply X and Y

always mean topological spaces on which no separation axioms are

assumed unless explicitly stated. Let A be a subset of a space X. The

closure of A and interior of A are denoted by cl ( A ) and int(A)

respectively. A subset A is said to be regular open (resp. regular closed)

if int . intA cl A resp A cl A . A subset A is said to be

-open [10] if AC int cl int (A). cl A denote the intersection of all

-closed sets containing A . A subset A of X is said to be g -closed

if cl A U whenever A U of U is -open. A subset A is said

g-Normal Spaces, Almost g-Normal … 3

to be a -neighbour hood of x if these exist a -open set U such that

x U A .

2.1 Definition: A subset A of a space ,X is called.

1) a generalized -closed (briefly g -closed) [7] if clA U and

whenever A U and U is -open.

2) a -generalized closed set (briefly g -closed) [8] if if clA U

and whenever A U and U is open.

3) a generalized preclosed set (briefly gp -closed) [1] if pclA U

whenever A U and U is open.

4) a pre open set [9] if intA clA .

5) S - relative to X [17] if every cover of A by semi-open sets of X has

a finite subfamily whose closures cover A .

6) g -open (resp g -open, gp -open) if the complement of A is g -

closed (resp. g -closed gp -closed)

2.2 Definition: A function : , ,f X Y is called

a) Completely continuous [2] if 1f V is regular open in X for every

open set V of Y .

b) rc-preserving [13] if f F is regular closed in Y for every regular

closed set F of X .

c) Almost closed [15] if f F is closed in Y for every regular closed

F of X .


d) Closed [5] if f F is closed in Y for every closed set F of X .

e) g -closed [7] (resp g -closed ) if f F is g -closed ( resp

g -closed ) for each closed set F of ,X .

f) -closed [12] if f F is -closed in Y for each closed set F in

,X .

g) M g -open if f F is g -open in Y for every g -open

set F in X.

h) g -irresolute if 1f V is g -closed in X for every

g -closed set V of Y.

2.3 Definition: A space X is said to be P -normal [11] if for any pair of

disjoint closed sets 1F and 2F , there exist disjoint pre open sets U and

V such that 1F U and 2F V .

3. g -Normal spaces

3.1 Definition: A space ,X is said to be g -normal if for any pair

of disjoint closed sets A and B these exist disjoint g -open sets U

and V such that A U and B V .

3.2 Theorem: Every normal space is g -normal.

Proof: Straight forward.


Converse of the above is not true as seen in the following example.

3.3 Example: Let , , ,X a b c d , , , , , , , , ,Z X b c d a c d c d ,

Then X is g -normal but it is not normal since the pair of closed sets

a and b have no disjoint neighbourhood.

Characterization of g -normality:

3.4 Theorem: For space X the following are equivalent

a) X is g -normal

b) For every pairs of open sets U and V whose union in X , there exist

g -closed sets A and B such that A U , B V and

A B X .

c) For every closed set F and every open set G containing F , these

exist a g -open set U such that F U cl U G .

Proof: ( a b ): Let U and V be a pair of open sets such that

X U V . Then X U X V . Since X is g -normal

there exist disjoint g -open sets 1U and 1V such that 1X U U and

1X V V . Let 1A X U and 1B X V . Then A and B are g -

closed sets such that A U , B V and A B X .

( b c ). Let F be a closed set and G be an open set containing F .

Then X F and G are open sets whose union is X . Then by (b) these

exist g -closed sets 1W and 2W such that 1W X F and 2W G


and 1 2W W X . Then 1F X W , 2X G X W and

1 2X W X W . Let 1U X W and 2V X W . Then U

and V are disjoint g -open set such

that F U X V G F U cl U X V G .

( c a ). Let A and B any two disjoint closed subsets of X . Then

A X B . Put G X B . Then G is an open set containing A . By

(c) there exist a g -open set U of X such that A U cl U G .It

follows that B X cl U . Let V X cl U . Then V is a

g -open set and U V . Therefore X is g -normal.

3.5 Theorem: A regular closed subspace of a P -normal space is

g -normal.

Proof: Let Y be a regular closed subspace of a P -normal space X . Let

A and B be two disjoint closed subsets of Y . Since Y is regular closed

it is closed in X . Therefore A and B are two disjoint closed subsets of

X . Since X is P -normal these exist pre-open sets U and V of X

such that A U and B V . As every regular closed set is semi-open,

Y is semi open, A U Y and B V Y U Y ,V Y are -open in

Y and hence g -open. Therefore Y is g -normal.

3.6 Definition: A function :f X Y is said to be almost- -irresolute if

for each x X and each nbd V of f x , 1cl f V is a

nbd of x .


3.7 Lemma: For a function :f X Y the following are equivalent

a) f is almost- -irresolute

b) 1 1intf V cl f V for every V O Y

Proof: Straight forward

3.8 Lemma: Let A be a subset of ,X and x X . Then x clA iff

V A for every -openset V containing x .


3.9 Lemma: If :f X Y is almost- -irresolute then

f clU cl f U for every U O X .

Proof: Let U O X . X suppose y cl f U . Then by lemma

3.8, these exist V O Y such that V f U . Hence

1f V U . Since U O X we have

1int cl f V clU . Since f is almost- -irresolute

1f V clU by lemma 3.7. Hence V f cl U implies

y f cl U . Therefore f clU cl f U

Invariance of g -normality


3.10 Theorem: If :f X Y is an M g -open continuous, almost-

-irresolute surjection from a g -normal space X on to a space Y ,

then Y is g -normal.

Proof: Let A be a closed subset of Y and B be an open set containing

A . Then by continuity of f , 1f A is closed and 1f B is an open

set of X such that 1f A 1f B . As X is g -normal, there

exist a g -open set U in X such that 1 1f A cl U f B

(By Theorem : 3.4 ). Then A f U f cl U B . Since f is

almost- -irresolute, M g -open, We obtain

A f U cl f U B . Again by theorem 3.4 Y is

g -normal.

3.11 Lemma: A mapping :f X Y is M g -closed iff for each

subset B in Y and for each g -open set U in X containing 1f B ,

there exist a g -open set V containing B such that 1f V U .

Proof: Necessity: Let :f X Y be M g -closed. Let B be an

open subset of Y and U be g -open set in X containing 1f B

such that 1f B U . Then Y f X U V is an g -open set

containing B such that 1f V U .


Sufficiency: Let F be a g -closed set in X .

1f Y f F X F . By taking B Y f F and U X F ,

these exist a g -open set V of Y containing B and 1f V U .

Then we have 1F X U X f V and Y V f F . Since

Y V is g -closed, f F is g -closed, and hence f is M g -

closed.

3.12 Theorem: If :f X Y be M g - closed continuous function

from a g -normal space on to a space Y , then Y is g -normal.

Proof: Let A and B be any disjoint closed sets of Y . Then 1f A and

1f B are disjoint closed sets of X . Since X is g -normal there

exist g -open sets U and V such that 1f A U and 1f B V .

By lemma 3.11 there exist g -open sets G and H of Y , such that

1, ,A G B H f G U and 1f H V . Since U and V are

disjoint, G and H are disjoint and hence Y is g -normal.

3.13 Theorem: If :f X Y is M g -closed map from a weakly

Hausdorff g -normal space X on to a space Y such that 1f y is

the S -closed relative to X for each y Y , then Y is g - 2T .


Proof: Let 1y and 2y be any two distinct point of Y . Since X is weakly

Hausdorff, 11f y and 1

2f y are disjoint closed such sets of X by

lemma 2.2[4]. As X is g -normal, there exist disjoint g -open

sets 1U and 2U containing 11f y and 1

2f y . Since f is

M g -closed, by lemma 3.11, there exist g -open sets 1V and 2V

in Y containing 1y and 2y , such that 1if V

iU for i =1,2. If follows

that 1 2V V . Hence the space Y is g - 2T .

3.14 Theorem: If :f X Y is an -closed continuous surjection and

X is normal then Y is g -normal.

Proof: Let A and B be disjoint closed sets of Y . Then 1f A , 1f B

are disjoint closed sets of X by continuity of f . As X is normal there

exist disjoint open sets U and V in X such that 1f A U

and 1f B V . By prop.(6) in [12], there are disjoint -open sets

G and H in Y such that A G and B H . Since every -open set is

g -open, G of H are disjoint g -open sests containing A and B

respectively. Therefore Y is g -normal.

3.15 Theorem: If :f X Y is continuous, g closed surjection and if

,X is normal then ,Y is g -normal.


Proof: Let A , B be disjoint closed sets of Y . Since X is normal there

exist disjoint open sets U and V of X such that 1f A U and

1f B V . By theorem 2.3.3 of [13] there exist g -open sets G and

H of ,Y such that A G , B H and 1f G U and 1f H V .

Then we have 1 1f G f H and hence G H . Since G is

g -open and A is -closed, A G implies intA G

Similarly intB H . Therefore int intG H G H and

hence Y is g -normal.

3.16 Theorem: If :f X Y is continuous, g -closed surjection from

a normal space ,X to ,Y then ,Y is g -normal.

Proof: Similar as theorem 3.15

3.17 Theorem: If :f X Y is g -irresolute, closed injection and

Y is g -normal then X is g -normal.

Proof: Let A and B be closed sets in X . Since f is closed injection,

f A and f B are disjoint closed sets of Y . Since Y is g -normal,

there exist g -open sets U and V such that f A U and

f B V such that U V .Since 1A f U and 1B f V and

f is g -irresolute, 1f U and 1f V are g -open sets in X such

that 1 1f U f V . Therefore X is g -normal.


4. Almost g -normal spaces

4.1 Definition: A space X is said to be almost g -normal if for each

closed set A and each regular closed set B such that A B , there

exist disjoint g -open sets U and V such that A U and B V .

4.2 Theorem: Every g -normal space is almost g -normal

Proof: Straight forward.

Converse of the above is not true as seen in the following example:

4.3 Example: Let , ,X a b c and , , , ,X a a b a c . X is

almost g -normal but not g -normal since b and c have no

disjoint g -open sets containing it.

Now, we have characterization of almost g -normality.

4.4 Theorem: For a space X , the following are equivalent

a) X is almost g -normal

b) For every pair of sets U and V one of which is open and the other is

regular open whose union is X , there exist g -closed sets G and

H such that ,G U H V and G H X .

c) For every closed set A and every regular open set B containing

A there is a g -open set V such that A V cl V B .


Proof: ( a b ): Let U be an open set and V be regular open set such

that U V X . Then X U X V . X U is closed and

X V is regular closed. Since X is almost g -normal, there exist

disjoint g -open sets 1 1,U V such that 1X U U and 1X V V .

Let 1G X U and 1H X V . Then G and H are g -closed sets

such ,G U H V and G H X .

( b c ) and ( c a ) are obvious.

Invariance of almost g -normality

4.5 Theorem: If :f X Y is continuous, M g open,

rc continuous and almost - -irresolute surjection from an almost

g -normal space X onto a space Y , then Y is almost g -normal.

Proof: Similar as theorem 3.10.

5. Mildly- g -Normal Spaces

5.1 Definition: A space X is said to be mildly g -normal if for every

pair of disjoint regular closed sets A and B of X , there exist disjoint

g -open sets U and V of X such that A U and B V .

5.2 Theorem: Every mildly normal space is mildly g -normal.



Converse of the above is not true as seen in the following example.

5.3 Example: , ,X a b c , , ,X a b a b . X is mildly

g -normal but not mildly normal since the regular closed set ,b c is

not contained in open set of X .

Characterization of mild g -normality.

5.4 Theorem: For space X the following are equivalent.

a) X is mildly g -normal

b) For every pair of regular open sets U and V whose union is X there

exist g -closed sets G and H such that ,G U H V and

G H X .

c) For any regular closed set A and every regular open set B containing

A there exists a g -open set U such that A U cl U B .

d) For every pair of disjoint regular closed sets there exist g -open

sets U and V such that ,A U B V and cl U cl V .


5.5 Theorem: If :f X Y is an M g open, rc continuous and

almost- -irresolute surjection from a mildly g -normal space X

onto a space Y then Y is mildly g -normal.


Proof: Let A be a regular closed set and B be a regular open set

containing A . Then by rc -continuity 1f A is a regular closed set

contained in the regular open set 1f B . Since X is mildly g -

normal there exist a g -open set V such that

1 1f A V cl V f B . As f is M g open and almost

g -irresolute surjection, it follows that A f V cl f V B .

Hence Y is mildly- g -normal.

5.6 Theorem: If :f X Y is rc continuous, M g closed map

from a mildly g -normal space X onto a space Y then Y is mildly

g -normal.


5.7 Theorem: If :f X Y is g -irresolute, rc preserving injection

and Y is mildly g -normal then X is mildly g -normal.

Proof: Let A and B be any disjoint regular closed sets of X . Since f

is rc preserving injection, f A and f B are disjoint regular closed

sets of Y . By mildly g -normality of Y , there exist g -open sets

U and V of Y such that f A U and f B V . 1 1,f U f V are

disjoint g -open sets containing A and B respectively. By theorem

5.4 X is mildly g -normal.


5.8 Theorem: If :f X Y is g -irresolute, almost closed injection

and Y is g -normal, then X is mildly g -normal.

Proof: Let A and B be disjoint regular closed sets in X . Since f is

almost closed injection, f A and f B are disjoint closed sets in Y .

Since Y is g -normal, there exist g -open sets U and V such that

f A U and f B V such that U V . Since f is g -

irresolute, 1 1,f V f U are g -open sets such that 1A f U

and 1B f V . By theorem 5.4. X is mildly- g -normal.

5.9 Theorem: If :f X Y is completely continuous, M g open

surjection and X mildly g -normal then Y is g -normal.

Proof: Let A and B be disjoint closed subsets of Y . Since f is

completely continuous, 1f A and 1f B are disjoint regular closed

subsets of X . X is mildly g -normal implies that there exist

g -open sets U and V in X such that 1f A U and 1f B V .

Since f is M g open surjection Y is g -normal.

6. Characterizations of almost -Normal and Mildly -Normal Spaces

6.1 Definition: A space X is said to be


a) -normal [3] if for every pair of disjoint closed sets A and B be of

X , there exist disjoint -open sets U and V such that A U and

B V .

b) Almost- -normal if for each closed set A and regular closed set of

B of X , such that A B there exist disjoint -open sets U and

V such that A U and B V .

c) Mildly- -normal if for every pair of disjoint regular closed sets

A and B of X , there exist disjoint -open sets U and V such that

A U and B V .

6.2 Lemma: A subset A of a space X is g - open (resp. g -open) iff

intF A whenever F is regular closed (resp. closed) and F A .

6.3 Theorem: The following are equivalent for a space X .

a) X is almost -normal.

b) For each closed set A and regular closed set B such that A B

there exist disjoint g -open sets U and V such that A U and

B V .

c) For each closed set A and regular closed set B such that A B

there exist disjoint g -open sets U and V such that A U and

B V .

d) For each closed set A and each regular open set B containing A ,

there exist a g -open set V of X such that A V clV B

Proof: ( a b c ) is obvious


( c d ): Let A be a closed set and B be a regular open subset of X

containing A. X-B is regular closed and by (c) there exist g - open

sets V and W such that A V and X B W . By

Lemma 6.2: intX B W and intV W . Therefore

intcl V W and hence intA V cl V X W B .

( d a ) Let A and B be closed and regular closed sets respectively.

Then X B is regular open set containing A . By (d) there exist a

g -open set G of X such that A G cl G X B put

intU G and V X cl G . Then U and V are disjoint -open

sets of X such that A U and B V . Hence X is almost -normal.

6.4 Theorem: The following are equivalent for a space X .

a) X is mildly -normal

b) For any disjoint regular closed sets A and B of X there exist

disjoint g -open sets U and V such that A U and B V .

c) For any disjoint regular closed sets A and B of X there exist

disjoint g -open sets U and V of X such that A U and

B V .

d) For each regular closed set A and each regular open set B

containing A there exist a g -open set V of X such that

A V cl V B .


e) For each regular closed set A and each regular open set B containing

A , there exist a g -open set V of X such that

A V cl V B .

Proof: Proof is similar to theorem 6.3.

References

[1]. AROCKIARANI I., BALACHANDRAN K. & DONTCHEV J"Some characterization of gp -irresolute and gp -continuous maps between topological spaces" Mem Fac Sci Kochi Univ Ser A(Math) 20 (1999) 93-104.

[2]. ARYA S. P. & GUPTA R. "On strongly continuous functions"Kyungpook Math J 14 (1974) 131-41.

[3]. DEVI R. "Studies on generalizations of closed maps and homeomorphisms in topological spaces" Ph.D. Thesis, Bharathiar University, Coimbatore (1994).

[4]. GARG G. L. & SIVARAJ D. "Pre semiclosed mappings", Periodica Math Hung., 19 (1988) 97-106.

[5]. JAMES R MUNKRES "Topology" second edition.

[6]. LEVINE N. "Generalized closed sets in topology"Rend.circ.mat.palermo (2), 19 (1970) 89-96.

[7]. MAKI H., DEVI R. & BALACHANDRAN K. Bull.Fukuoka Univ Ed.part III 42 (1993),13-21.

[8]. MAKI H., DEVI R. & BALACHANDRAN K.Mem.Fac.Kochi.Univ.Ser.A Math 15 (1994), 51-63.

[9]. MASHHOUR A. S., ABD EL-MONSEF M E, & EL-DEEB S. N. "On pre-continuous and weak pre-continuous functions"proc.Math phy soc Egypt 53 (1985) 47-53.


[10]. MASHHOUR A. S., HASANEIN I. A. & EL.DEEB S. N. " -continuous and -open mappings", Acta Math.Hung. 41(1983), 213-218.

[11]. NAVALGI G. B. " p -normal, almost- p -normal and mildly p -normal spaces", Topology Atlas preprint 427.URL: http:

//at.york u.ca/i/d/e/b/71.html

[12]. NOIRI T. "Almost continuity and some separation axioms"Glasnik Mat., 9(29) (1974), 131-135.

[13]. NOIRI T. "Mildly normal spaces and some functions" Kyungpook Math J 36 (1996) 183-190.

[14]. NOUR T. M. "Contributions to the theory of bitopological spaces"Ph.D thesis, Delhi University, India 1989.

[15]. SINGAL M. K. & SINGAL A. R. "Almost continuous Mappings"Yokohama Math J 16 (1968) 63-73.

[16]. SINGAL M. K. & SINGAL A. R. "Mildly normal spaces"Kyungpook Math J 13 (1973) 27-31.

[17]. THOMPSON T. "S-closed spaces", PAMS 60 (1976) 335-338.



University – 2013

THE BEHAVIOR OF SOME MODULES IN ][M

OVER AN HNP RING

IrawatiAlgebra Group, Faculty of Mathematics and Natural Sciences,

Institut Teknologi Bandung, Jalan Ganesha no 10Bandung, INDONESIA

(Communicated by Prof. Dr. Stefan Veldsman)

Abstract

For a finitely generated module M over an HNP ring, we obtain the behavior of some modules in ][M , where M is a module over an HNP ring. We also obtain a characterization of an HNP ring

Introduction

It is well known that a ring is called hereditary if all of its left and

right ideals are projective, it is called a Noetherian if the ideals are

finitely generated and is prime if the annihilator of the ideals is {0}.

McConell and Robson have already proven that a finitely generated

module over an HNP ring can be decomposed into a direct sum of a

projective module and a torsion module [5]. They also proved that a

finitely generated module over an HNP ring is a torsion module if and

only if its length is finite [6]. Levy has proven that a finite length module

over an HNP ring can be decomposed into a direct sum of a module

annihilated by an invertible ideal and a module that has no composition

factors annihilated by an invertible ideal [4]. Wisbauer, in [7],

introduced the full subcategory ][M of the category R-Mod, as the

2 Irawati

category of all submodules of a module which is a homomorphic image

of a tupple of M. In this paper we call all modules in ][M as modules

in the neighborhood of M. We investigate the behavior of modules in the

neighborhood of a finitely generated projective module M over an HNP

ring, the behavior of modules in the neighborhood of a finite length

module M annihilated by an invertible ideal over an HNP ring, and the

behavior of modules in the neighborhood of a finite length module M

over an HNP ring that has no composition factors annihilated by an

invertible ideal. In this paper all rings have a unit element.

1. Finitely Generated Projective Modules over an HNP Ring

The definition and the characterization of a prime ring, can be

found in [6]. We already know that a ring R is called prime if for every

non zero element a and b, there is a non zero element r in R such that

arb is not zero. It is proved in [6] that a ring is prime if and only if the

annihilator of every non zero right and left ideal is zero. Based on that

characterization of the prime ring, the concept of prime modules is

introduced in [1], [2] and [3], that a module is called prime if the

annihilator of every non zero submodule is the same as the annihilator of

the module itself. We call a hereditary, Noetherian, prime module as a

HNP module.

The first result, is the characterization of an HNP ring. This result

completes the table of rings characterized in [8].

The Behavior of Some Modules in [M] over an HNP Ring 3

1.1 Theorem: A ring R is an HNP ring if and only if every finitely

generated right projective R-module is an HNP module.

Proof: Let M be a finitely generated projective right R-module. Let

RM , for a finite set . From a proposition in [7], we have

ii KM , with iK is a right ideal in R. Because R is a prime ring,

then we have that iKann i 0)( . So 0)( Mann . It can be seen

that the annihilator of every submodule of M is also zero. Then we can

conclude that M is a prime module. Every submodule N of M is in the

form iLN where iL is are right ideals that are contained in iK . So

obviously N is finitely generated and it is projective since the direct sum

of projective ideals is also projective. This means that M is an HNP

module.

Because R, as a module over itself, is a projective module and is

finitely generated, then R is an HNP module as a right module over

itself. In other words, R is an HNP ring.

In the following, R is an HNP ring. From the proof of Theorem 1.1, it is

seen that for a finitely generated projective R-module M, then

0)( Mann . In the next Theorem, for a finitely generated projective

R-module M, we will prove that every projective module in ][M also

has zero annihilator. Here ][M is a subcategory of R-mod, consists of

submodules of a module which is a homomorphic image of M , where

is an index set [7].

4 Irawati

1.2 Theorem: Let M be a finitely generated right projective R – module,

and P is a projective module in ][M . Then 0)( Pann .

Proof: According to the proof of Theorem 1.1, we have that

0)( Mann and M an HNP module. Because of ][MP , there is an

epimorphism ': PM with 'PP . Because P is projective in

][M , then we have MPP )(1 . So ii MP , where iM is

a submodule of M. Because M is a prime module, then

0)()( MannMann i . Thus 0)( Pann .

Next we will see that the direct sum of HNP R-modules is also an HNP

R-module.

1.3 Theorem: Let M be a finitely generated right projective R – module,

and PP with P a family of HNP modules in ][M , for a

finite set . Then P is an HNP module in ][M .

Proof: It is seen in [7] that P is hereditary in ][M . So according to

Theorem 1.2, we have that 0)( Pann . Let Q be a non zero submodule

of P. Then ii QQ with iQ is a submodule of iP . Because iQ is

projective in ][M , according to Theorem 1.2, 0)( iQann . So

0)()( QannPann . Every submodule N of P is in the form

iLN where iL is are submodules that are contained in iP . So

obviously N is finitely generated and it is projective since the direct sum


of projective modules is also projective. We have P is an HNP module in

][M .

Now we will characterize an HNP module.

1.4 Theorem: Let M be a finitely generated projective R-module, and P

a projective module in ][M . Then the following are equivalent.

1. P is an HNP module in ][M .

2. For every finite index set , )(P is an HNP module

3. Every finitely subgenerated projective module in ][P is an HNP

module in ][M .

Proof: ( 21 ) It is obvious from Theorem 1.3.

( 32 )Let K be a finitely subgenerated projective module in ][P .

There is an epimorphism ': 11 KP for a finite index set 1 and K

is a submodule of 'K . We have the following exact sequence in ][P ,

0)(11 KK . So K is isomorphic to a submodule of

1)(1 PK , because K is a projective module in ][P . Because

1P is an HNP module in ][M , the module K is also an HNP module in

][M .

( 13 ) Let ][PA . Then there is an epimorphism ': 11 AP with

'AA . From the hypotheses we know that ][MP . So there is an

epimorphism ': 22 PM , with '1 PP . We have the following

6 Irawati

epimorphism ')(: 11212 AP . Because )( 11

2 P is a submodule

of 2M , we have ][)( 112 MP . And 'A , as a homomorphic image

of )( 112

P , is also in ][M . So P is also projective in ][P , and

hence P is an HNP module in ][M .

Let P be a finitely generated projective R-module. From Theorem 1.1,

we have that P is an HNP module. From Theorem 1.4, by taking R = M,

we can conclude that every finitely subgenerated projective module in

][P is an HNP module. It is evident that we can see the behavior of

some modules in the neighborhood of a finitely generated projective

module over an HNP ring. So we get the characterization of an HNP

R-module P, that P is a projective module and every finitely

subgenerated projective module in ][P is also an HNP module.

2. Finitely Generated Torsion Modules over an HNP Ring

In this section, R is an HNP ring.

2.1 Finite length module annihilated by an invertible ideal

The next Theorem is a characterization of a finite length R-module that is

annihilated by an invertible ideal.

2.1.1 Theorem: Let M be a finite length R-module. Then the following

are equivalent:


1. M is annihilated by an invertible ideal.

2. Every R-module in ][M is annihilated by an invertible ideal

3. Every simple submodule of M is annihilated by an invertible ideal

Proof: ( 21 ) Let M be annihilated by an invertible ideal I. Let

][MN , and N a submodule of 'N , the homomorphic image of

': NM . So 'N is also annihilated by I. Thus N is also annihilated

by I.

( 32 ) The simple submodule of M is also in ][M .

( 13 ) Let 1S be a simple submodule of M that is annihilated by an

invertible ideal 1I . Then 11 MMI is a proper submodule of M. Let

2S be a simple submodule of 1M that is annihilated by an invertible ideal

2I . Then 21IM is a proper submodule of 1M . We continue the process

until we have that 021

21 ktk

tt IIIM . This will be happened

because iI is an invertible ideal for every ki ,,1 , and the product of

invertible ideals is also an invertible ideal. So M is annihilated by an

invertible ideal ktk

tt III 21

21 .

2.2. Finite length module that has no composition factors annihilated

by an invertible ideal.

In this part we characterize a finite length R – module that has no

composition factors annihilated by an invertible ideal.

8 Irawati

First we characterize a finite length R – module that has no composition

factors annihilated by an invertible ideal, using its simple submodules.

At first, we state Lemma 4.3(i) in [4].

2.2.1 Lemma: Let S,T be simple R-modules. If there is an invertible

ideal annihilating S but not annihilating T, then Ext1(S,T) = 0

In the next Theorem, we characterize a finite length R-module

that is annihilated by an invertible ideal.

2.2.2 Theorem: Let M be a finite length R-module. Then M is

annihilated by an invertible ideal if and only if each of its simple

submodules is annihilated by an invertible ideal.

Proof: We prove by induction on the composition length of M. Let S be

a simple module of M annihilated by an invertible ideal I, and let T be a

simple submodule of M/S. If Ext1(T,S) = 0, then T is isomorphic to a

submodule of M and hence is annihilated by an invertible ideal. If

Ext1(T,S) is non zero, then by Lemma 2.2.1, T is annihilated by the same

invertible ideal I as is S. Either way, the hypotheses of the Theorem are

satisfied by M/S, so by induction it is annihilated by some invertible

ideal J, and then M is annihilated by JI.

As consequences we have the following


2.2.3 Corollary: Let M be a finite length R-module not annihilated by

invertible ideals. Then M has a simple submodule not annihilated by

invertible ideals.

Next we will see a Theorem that is related to the extension of a finite

length R-module by a finite length R-module.

2.2.4 Theorem: Let T, M be finite length R-modules. Let Ext1(T,X) = 0

for every composition factor X of M. Then Ext1(T,M) = 0.

Proof: The proof is by induction on the composition length of M. The

case of length 1 being trivial. Let see the following exact sequence

Ext1(T,S) Ext1(T,M) Ext1(T,M/S)

Note that Ext1(T,M/S) = 0 by induction, and Ext1(T,S) = 0 by assumption,

so Ext1(M ,S) = 0.

Similarly we have the next Theorem

2.2.5 Theorem: Let T, M be finite length R-modules. Let Ext1(X,T) = 0

for every composition factor X of M. Then Ext1(M,T) = 0.

In the next Theorem we characterize a finite length module that has no

composition factors is annihilated by invertible ideals.

10 Irawati

2.2.6 Theorem: Let M be a finite length R-module. Then M has no

composition factors is annihilated by invertible ideals if and only if each

of its simple submodules is not annihilated by invertible ideals.

Proof: Suppose that each of the simple submodules of M is not

annihilated by invertible ideals. The Theorem will be proved by

induction on the length. Let the proposition be true for modules with

length < n. It will be proved that it is true for modules with length n. Let

M1 be a submodule of M with length n – 1. Then M1 only has simple

submodules that are not annihilated by invertible ideals. For the next

composition series 0 S = Mn-1 ……. M2 M1 M,

The factor module M1/S has no composition factors annihilated by

invertible ideals. It will be proved that M/M1 is not annihilated by

invertible ideals. We have that M/M1 ≈ (M/S)/(M1/S).

Let see the next exact sequence

0 M1/S M/S (M/S)/(M1/S).

Suppose (M/S)/(M1/S), annihilated by an invertibe ideal. By Lemma

2.2.1, Ext((M/S)/(M1/S),T) = 0 for every composition factor T of M1/S.

By Theorem 2.2.4, then Ext((M/S)/(M1/S), M1/S) = 0. It means that the

exact sequence above is split. So we have (M/S)/(M1/S) is isomorphic to

a simple submodule of M/S, say K/S. So K/S is a composition factor of a

submodule of M with length n – 1. So K/S is not annihilated by an

invertible ideal. This is contradicting with the fact that (M/S)/(M1/S) is

annihilated by an invertible ideal. So we have that M/M1 is not

annihilated by an invertible ideals. The converse is obviously true.


Now we will prove a corollary of the above theorem that characterizes a

finite length R – module M that has no composition factors annihilated

by an invertible ideal via the same property of finitely subgenerated

modules in ][M .

2.2.7 Corollary: An R-module M has no composition factors annihilated

by an invertible ideal if and only if every finitely subgenerated module in

][M has no composition factors annihilated by an invertible ideal.

Proof: ( ): It is obvious because M is a finitely subgenerated module in

][M .

( ): First we will show that M , for a finite set , has no

composition factors annihilated by an invertible ideal. Let K be a simple

submodule of M . We construct the following submodule of M :

aaKaMaM jij

k , .

Suppose K is annihilated by an invertible ideal I. Then jkM will also

annihilated by I. This means that M has a composition factor annihilated

by an invertible ideal. This is contradicts the statement that M has no

composition factors annihilated by an invertible ideal. So M , for a

finite set , has no composition factors annihilated by an invertible

ideal. Now let L be a finitely subgenerated module in ][M , and "L a

simple submodule of L. Then there is a finite set , and an epimorphism

': LM , with L a submodule of 'L . For the

following epimorphism, "":)"(/' 11 LLL , then

12 Irawati

)'(/)"(" 1 KerLL . On the other hand, )'(/)"(1 KerL is a

composition factor of M . Because M has no composition factors

annihilated by an invertible ideal, then )'(/)"(1 KerL is not

annihilated by an invertible ideal. So "L is not annihilated by an

invertible ideal. According to Theorem 2.2.6, L has no composition

factors annihilated by an invertible ideal.

References

[1]. DESALE G. B. & VARADARAJAN K. "SP modules and related topics", Research paper 463 (1980) Calgary, Alberta, Canada.

[2]. GOODEARL K. R., HANDELMAN D. & LAWRENCE J. "Strongly prime and completely torsion free rings", Carleton Mathematical Series 109 (1974).

[3]. JOHNSON R. E. "Representations of Prime Rings", Transactions of the AMS, Vol 74(1) (1953), 351-357.

[4]. KLINGLER L. & LEVY L. S. "Wild torsion modules over Weyl algebras and general torsion modules over HNPs", Journal of algebra 172 (1995), 273-300.

[5]. Mc CONNELL J. C. & ROBSON J. C. "Homomorphisms and extensions of modules over certain differential polynomial rings", Journal of Algebra 26 (1973), 319-342.

[6]. Mc CONNELL J. C. & ROBSON J. C., "Noncommutative Noetherian Rings", John Wiley (1987).

[7]. WISBAUER R. "Foundations of module and ring theory", Gordon and Breach (1991).

[8]. WISBAUER R., "Module and Comodule Categories-a Survey", Proceedings of the Mathematics Conference, Birzeit Conf 1998, Elyadi e.a., World Scientific, 277-304 (1999).



University – 2014

(Communicated by Prof. Dr. K. Sitaram, SVU)

2 Alagar

Generalized Closed Sets with respect to an Ideal 3

4 Alagar

6 Alagar

AUTHORS BIO-DATA

Dr. C. Janaki, is now working as an Assistant Professor in LRG Government arts College for Women, Tiruppur-4. Awarded Ph.D by Bharathiar University in 2010. She is having 17 years of teaching experience. e-mail: [email protected]

Dr. Sr. I. Arochiarani, working as AssociateProfessor of Mathematics, Nirmala College for Women, Red Fields, Coimbatore-641018. Awarded

Ph.D., degree by Bharathiyar University in 1998. 4 Ph.D’s and 20 M.Phil.,’s were awarded under her guidance. Published 40 research papers in National and International Journals.

Dr Nand Lal Singh is working as Professor and Head of the Department of Mathematics, T.D.P.GCollege, Jaunpur, Utter Pradesh. He is having 42 years of teaching experience of Post graduate classes and published 45 research papers in International & National reputed journals.

Dr Jitesh Kumar Singh is working as a Lecturer in Mathematics, T.D.P.G college, Jaunpur, Utter Pradesh. He is having 6 years of teaching experience of Post graduate classes and published 15 research papers in different journals of repute.

ISSN 0973-3477_____________________________________________

Acharya Nagarjuna InternationalJournal of Mathematics

and Information Technologye-mail: [email protected]

Vol: 7 ∙ 2010 , Vol: 8 ∙ 2011, Vol: 9 ∙ 2012, Vol: 10 ∙ 2013, Vol: 11 . 2014

CONTENTSResearch Papers Page No

Volume ∙7

Viscous Dissipation of Energy due to slow motion of Porous 01-13 Newtonian Sphere in Visco-Elastic Fluid.

Jitesh Kumar Singh & Nand Lal Singh

Volume ∙8A Note on Weighted Ostrowski – Gruss type inequality 01 - 10 and Applications.

A. Qayyum

Volume ∙09g -Normal Spaces Almost g -Normal And 01-20

Mildly g -Normal Spaces

C. Janaki & Sr. I. Arockiarani

Volume ∙10

. The Behavior of Some Modules in ][M 01-12 Over an HNP Ring

IrawatiVolume ∙11. Generalized Closed Sets with Respect to an Ideal 01-07 R. Algar

ANIJMIT VOL 7-11 2010-2014

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Transcript of ANIJMIT VOL 7-11 2010-2014