Research Article The Generalization of Prime...

Hindawi Publishing CorporationAlgebraVolume 2013 Article ID 581023 4 pageshttpdxdoiorg1011552013581023

Research ArticleThe Generalization of Prime Modules

M Gurabi

Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156-83111 Iran

Correspondence should be addressed to M Gurabi m gurabimathiutacir

Received 29 December 2012 Accepted 15 February 2013

Academic Editor Masoud Hajarian

Copyright copy 2013 M Gurabi This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Piecewise prime (PWP) module 119872119877is defined in terms of a set of triangulating idempotents in 119877 The class of PWP modules

properly contains the class of prime modules Some properties of these modules are investigated here

1 Introduction

All rings are associative and 119877 denotes a ring with unity 1The word ideal without the adjective right or left means two-sided ideal The right annihilator of ideals of 119877 is denoted byrann119877(119868) A ring 119877 is 119902119906119886119904119894-119861119886119890119903 (119861119886119890119903) if the right annihila-

tor of every right ideal (nonempty subset) of119877 is generated asa right ideal by an idempotentWenow recall a fewdefinitionsand results from [1] which motivated our study and serve asthe backgroundmaterial for the presentwork An idempotent119890 isin 119877 is a left semicentral idempotent if 119890119909119890 = 119909119890 for all 119909 isin 119877Similarly right semicentral idempotent can be defined Theset of all left (right) semicentral idempotents of 119877 is denotedby 119878119897(119877)(119878119903(119877)) An idempotent 119890 isin 119877 is semicentral reduced

if 119878119897(119890119877119890) = 0 119890 If 1 is semicentral reduced then 119877 is called

semicentral reduced An ordered set 1198901 119890

119899 of nonzero

distinct idempotents of 119877 is called a set of left triangulatingidempotents of 119877 if all the following hold

(i) 1198901+ sdot sdot sdot + 119890

119899= 1

(ii) 1198901isin 119878119897(119877)

(iii) 119890119896+1

isin 119878119897(119888119896119877119888119896) where 119888

119896= 1 minus (119890

1+ sdot sdot sdot + 119890

119896) for

1 le 119896 le 119899

From part (iii) of the previous definition it can be seen thata set of left triangulating idempotents is a set of pairwiseorthogonal idempotents A set 119864 = 119890

1 119890

119899 of left trian-

gulating idempotents of119877 is complete if each 119890119894is semicentral

reduced A (complete) set of right triangulating idempotentsis defined similarly The cardinalities of complete sets of lefttriangulating idempotents of 119877 are the same and are denotedby 120591 dim(119877) [1 Theorem 210] A ring 119877 is called piecewise

prime if there exists a complete set of left triangulatingidempotents 119864 = 119890

1 119890

119899 of 119877 such that 119909119877119910 = 0 implies

119909 = 0 or 119910 = 0 where 119909 isin 119890119894119877119890119895and 119910 isin 119890

119895119877119890119896for

1 le 119894 119895 119896 le 119899 In view of this definition we say a properideal 119868 in 119877 is a PWP ideal if there is a complete set of lefttriangulating idempotents 119864 = 119890

1 119890

119899 such that 119909119877119910 sube 119868

implies 119909 isin 119868 or 119910 isin 119868 where 119909 isin 119890119894119877119890119895and 119910 isin 119890

119895119877119890119896

for 1 le 119894 119895 119896 le 119899 If 119877 is PWP then it is PWP with respectto any complete set of left triangulating idempotents of 119877furthermore for a ring 119877 with finite 120591 dim(119877) 119877 is PWP ifand only if 119877 is quasi-Baer [1 Theorem 411]

A nonzero right 119877-module119872 is called a prime module iffor any nonzero submodule119873 of119872 rann

119877(119873) = rann

119877(119872)

and a proper submodule 119875 of119872 is a prime submodule of119872if the quotient module119872119875 is a primemoduleThe notion ofprime submodule was first introduced in [2 3] see also [4 5]It is easy to see that119872 is a prime 119877-module if and only if forany119898 isin 119872 and 119887 isin 119877 if119898119877119887 = 0 then119898 = 0 or119872119887 = 0

In this work the concept of prime modules is developedto piecewise prime modules as it is done for rings in [1]Throughout this work it is considered that 120591 dim(119877) is finite

2 Main Results

Definition 1 Let119872 be an 119877-module and 119878 = End119877(119872)

(1) 119872 is a piecewise prime (PWP) 119877-module with respectto a complete set of left triangulating idempotents119864 =1198901 119890

119899 of 119877 if for any119898 isin 119872 119890

119894isin 119864 and 119887 isin 119877

119898119890119894119877119890119894119887 = 0 997904rArr 119898119890

119894= 0 or 119872119890

119894119887 = 0 (1)

2 Algebra

(2) Let 119873 be a submodule of 119872 Then 119873 is a piecewiseprime submodule of119872 with respect to 119864 if119872119873 is aPWP module with respect to 119864

(3) 119872 is piecewise endoprime (PWEP) with respect to acomplete set of left triangulating idempotents 119865 =

1198871 119887

119898 of 119878 such that for each nonzero submod-

ule 119873 sube 119872 119891 isin 119878 and 119887119894isin 119865 if 119891119887

119894119873 = 0 then

119891119887119894= 0

By Definition 1 119873 is a piecewise prime submodule of119872with respect to a set of left triangulating idempotents 119864 if forany119898 isin 119872 119890

119894isin 119864 and 119887 isin 119877

119898119890119894119877119890119894119887 sube 119873 997904rArr 119898119890

119894isin 119873 or 119872119890

119894119887 sube 119873 (2)

Example 2 Let 119864 = 1198901 119890

119899 be a complete set of left

triangulating idempotents o 119877

(1) Let 1198961and 119896

2be two fields and 119877 = 119896

1times 1198962 Then

119872 = 1198771198961oplus 119877119896

2is not a prime module but it is

piecewise prime with respect to (1 0) (0 1)(2) If 119872 is a prime 119877-module then it is piecewise

prime with respect to any set of left triangulatingidempotents of 119877

(3) Homomorphic image of 119872119877needs to be PWP with

respect to 119864 For example ZZ is a PWP modulewith respect to 0 1 but Z

4is not PWP because

rann119903(2) = rann

119903(Z4)

Corollary 3 If119872 is a PWP 119877-module with respect to 119864 thenany submodule of119872 is PWP with respect to 119864

Proof It can be seen by Definition 1

Proposition 4 Let 119877 be a ring with finite triangulatingdimension

(1) 119868 is a PWP ideal of 119877 if and only if 119877119868 is a PWP 119877-module

(2) 119877 is a PWP ring if and only if 119877119877is PWP

Proof The part one is obtained by Definition 1 and forsecond let 119868 = 0 in part one

Proposition5 Let119872 be an119877-module and let119864 = 1198901 119890

119899

be a set of left triangulating idempotents of 119877 Then thefollowing statements are equivalent

(1) 119872 is PWP with respect to 119864(2) for each 119873 sube 119872 ideal 119868 in 119877 and 119890

119894isin 119864 if 119873119890

119894119868 = 0

then119873119890119894= 0 or119872119890

119894119868 = 0

(3) for each (119898) sube 119872 ideal (119886) in 119877 and 119890119894isin 119864 if

(119898)119890119894(119886) = 0 then (119898)119890

119894= 0 or119872119890

119894(119886) = 0

Proof (1) rArr (2) If 119873119890119894= 0 then there exists 119899 isin 119873 such

that 119899119890119894= 0 and for any 119887 isin 119868 119899119890

119894119877119890119894119887 = 0 By Definition 1

for each 119887 isin 119868119872119890119894119887 = 0 This implies that119872119890

119894119868 = 0

(2) rArr (3) In (2) let119873 = (119898) and 119868 = (119886)(3) rArr (1) Let 119898119890

119894119877119890119894119887 = 0 where 119898 isin 119872 119890

119894isin 119864 and

119887 isin 119877 Thus 119898119890119894119877119890119894119877119890119894119887119877 = 0 or (119898119890

119894)119890119894(119890119894119887) = 0 By (3)

(119898119890119894)119890119894= 0 or 119872119890

119894(119890119894119887) = 0 This implies that 119898119890

119894= 0 or

119872119890119894119887 = 0

Proposition 6 Let119872 be an 119877-module 119878 = 119864119899119889119877(119872) let 119864 =

1198901 119890

119899 be a complete set of left triangulating idempotents of

119877 and let119865 = 1198871 119887

119898 be a complete set of left triangulating

idempotents of 119878

(1) 119872 is a PWP 119877-module with respect to 119864 if and only iffor each119873 sube 119872with119873119890

119894= 0 119886119899119899

119903(119873119890119894) = 119886119899119899

119903(119872119890119894)

(2) If 119872119877

is PWP 119877-module with respect to 119864 then119886119899119899119903(119872) is a PWP ideal of 119877 with respect to 119864

(3) If119872119877is PWEP with respect to 119865 and retractable then

119886119899119899119903(119872) is a PWP ideal of 119877 with respect to 119864

Proof (1) If 119887 isin ann119903(119873119890119894) then there exists 119899 isin 119873 such that

119899119890119894119877119890119894= 0 and 119899119890

119894119877119890119894119887 = 0 Since119872 is PWP 119877-module with

respect to 119864 by Definition 1 119872119890119894119887 = 0 Hence ann

119903(119873119890119894) =

ann119903(119872119890119894) Conversely let119898119890

119894119877119890119894119887 = 0 where 119890

119894isin 119864119898 isin 119872

119887 isin 119877 and 119898119890119894= 0 Thus 119887 isin ann

119903((119898119890119894119877)119890119894) which means

119887 isin ann119903(119872119890119894) or119872119890

119894119887 = 0

(2) Let 119868119890119894119869 sube ann

119903(119872) and 119868119890

119894sube ann

119903(119872) Since

(119872119868119890119894)119890119894119869 = 0 and119872 is a PWP 119877-module with respect to 119864

by Proposition 5119872119890119894119869 = 0Thus 119890

119894119869 sube ann

119903(119872)This implies

that ann119903(119872) is a PWP ideal of 119877 with respect to 119864

(3) Let 119868119890119894119869 sube ann

119903(119872) where 119868119890

119894 119890119894119869 sube ann

119903(119872) Since

119872 is retractable then there exists a nonzero homomorphism119891 119872 rarr 119872119868119890

119894 There exists 119887

119895isin 119865 such that 119891119887

119895= 0

Since 119868119890119894119869 sube ann

119903(119872) 119891119887

119895119872119890119894119869 = 0 By assumption 119872 is

PWEP with respect to 119865 This implies that 119891119887119895= 0 which

is a contradiction Hence ann119903(119872) is a PWP ideal of 119877 with

respect to 119864

A module 119872119877is called retractable if for any nonzero

submodule119873 of119872 Hom119877(119872119873) = 0

Theorem 7 Let 119872 be an 119877-module 119878 = 119864119899119889119877(119872) and

let 119865 = 1198871 119887



(1) If119878119872 is a PWP module with respect to 119865 then 119878 is a

PWP ringThe converse is true when119872119877is retractable

(2)119878119872 is a PWP module with respect to 119865 if and only if119872119877is PWEP with respect to 119865

Proof (1) Let 119891119887119894119878119887119894119892 = 0 where 119891 119892 isin 119878 119887

119894isin 119865 and 119887

119894119892 = 0

Thus there exists119898 isin 119872 such that 119887119894119892119898 = 0 and 119891119887

119894119878119887119894119892119898 =

0 Since119878119872 is PWPwith respect to 119865119891119887

119894119872 = 0whichmeans

119891119887119894= 0 Conversely let 119891119887

119894119878119887119894119898 = 0 and 119887

119894119898 = 0 Since 119872

119877

is retractable there exists a nonzero homomorphism 119887119894119892 isin

Hom119877(119872 119887119894119898119877) Thus 119891119887

119894119878119887119894119892 = 0 Since 119878 is PWP 119891119887

119894= 0

(2) Assume119872 is a PWP 119878-module with respect to 119865 Let119873 sube 119872 and 119891119887

119894119873 = 0 where 119891 isin 119878 and 119887

119894isin 119865 Since

119878119872 is PWP by Proposition 6(1) 119891119887

119894119872 = 0 Thus 119891119887

119894=

0 Conversely assume 119872119877be PWEP with respect to 119865 Let

119891119887119894119878119887119894119898 = 0 where 119891 isin 119878 119887

119894isin 119865 119898 isin 119872 and 119887

119894119898 = 0

If 119873 = 119878119887119894119898 then 119891119887

119894119873 = 0 This implies that 119891119887

119894= 0 or

119891119887119894119872 = 0 Hence

119878119872 is PWP with respect to 119865

Algebra 3

Let119872 be a right 119877-module with 119878 = End119877(119872) Then119872

119877

is called a quasi-Baer module if for any119873sube119878119872 lann

119878(119873) =

119878119890 where 119890 = 1198902 isin 119878 [6]

Corollary 8 Let119872 be a retractable119877-module 119878 = 119864119899119889119877(119872)

and let 119865 = 1198871 119887


idempotents of 119878 Then the following statements are equivalent

(1) 119872119877is a PWEP module with respect to 119865

(2)119878119872 is a PWP module with respect to 119865

(3) 119872119877is quasi-Baer

Proof (1) hArr (2)This is evident byTheorem 7(2)(2) hArr (3) By [6 Proposition 47] 119872

119877is quasi-Baer if

and only if 119878 is quasi-Baer By [1 Theorem 411] 119878 is PWPwith respect to 119865 if and only if 119878 is quasi-Baer The result isobtained byTheorem 7(1)

Proposition 9 Let Λ be an index set and let 119864 = 1198901 119890

119899

be a complete set of left triangulating idempotents of 119877

(1) Let119872 = oplus120582isinΛ

119872120582119872 is PWP with respect to 119864 if and

only if for each 120582 isin Λ119872120582is PWP with respect to 119864

(2) Let119872 = prod120582isinΛ



Proof (1) Assume 119872 is PWP with respect to 119864 If119898120582119890119894119877119890119894119887 = 0 where 119898

120582isin 119872

120582 119890119894isin 119864 and 119887 isin

119877 then (0 119898120582 0 0)119890

119894119877119890119894119887 = 0 Since 119872 is PWP

(0 119898120582 0 0)119890

119894= 0 or 119872119890

119894119887 = 0 This implies

that 119898120582119890119894= 0 or 119872

120582119890119894119887 = 0 which means for each

120582 isin Λ 119872120582is PWP with respect to 119864 Conversely assume

that for each 120582 isin Λ 119872120582is PWP with respect to 119864 and

(1198981 119898

119899 0 )119890

119894119877119890119894119887 = 0 This implies that119898

120582119890119894119877119890119894119887 = 0

Since119872120582is PWP with respect to 119864 119898

120582119890119894= 0 or119872119890

119894119887 = 0

Hence (1198981 119898

119899 0 )119890

119894= 0 or119872119890

119894119887 = 0 Thus119872 is PWP

with respect to 119864(2) It can be seen by similar method as in part (1)

Corollary 10 Let 119864 = 1198901 119890


triangulating idempotents of 119877 let119872 be an 119877-module and let119865 be a free 119877-module

(1) 119877 is quasi-Baer if and only if 119865 is a PWP module withrespect to 119864

(2) 119872 is PWP with respect to 119864 if and only if 119865otimes119877M is

PWP with respect to 119864

Proof It follows by [1 Theorem 411] and Proposition 9

Proposition 11 Let 119872 be an 119877-module and 119878 = 119864119899119889119877(119872)

Then119878119872 is prime if and only if 120591 119889119894119898(119878) = 1 and119872

119877is quasi-

Baer

Proof (rArr) Since119872 is a prime 119878-module then for each119873 sube

119872 lann119878(119873) = lann

119878(119872) = 0This implies that119872

119877is quasi-

Baer If 1198902 = 119890 isin 119878 then119872 = 119890119872oplus(1minus119890)119872 Since119878119872 is prime

lann119878(119890119872) = lann

119878((1minus119890)119872) = lann

119878(119872)This implies that

119890 = 1 or 119890 = 0 Thus 120591 dim(119878) = 1

(lArr) Let 119873 be any submodule of119878119872 Since119872

119877is quasi-

Baer lann119878(119873) = 119878119890 where 119890 isin 119878

119903(119878) Since 120591 dim(119878) = 1 119890 isin

0 1 If 119890 = 1 then119873 = 0 Thus 119890 = 0 This implies that foreach nonzero submodule 119873sube

119878119872 lann

119878(119873) = lann

119878(119872) =

0 This means119878119872 is prime

It is folklore that prime radical plays an important role inthe study of rings [7] Following this concept is developed formodules of course by using a complete set of left triangulatingidempotents of 119877

Definition 12 Let 119872 be an 119877-module let 119873 be a propersubmodule of 119872 and let 119864 = 119890

1 119890

119899 be a complete set

of left triangulating idempotents of 119877

(1) The piecewise prime radical of 119873 in 119872 with respectto 119864 is denoted by PRad(119873) and is defined to be theintersection of all piecewise prime submodules of119872with respect to 119864 containing119873

(2) PRad(119872) means the intersection of all piecewiseprime submodules of 119872 with respect to 119864 If 119872 hasno piecewise prime submodulewith respect to119864 thenPRad(119872) = 119872

Proposition 13 Let119873 be a submodule of 119877-module119872

(1) If119873 is a submodule of 119877-module119872 then 119875119877119886119889(119873) sube119875119877119886119889(119872)

(2) If 119875119877119886119889(119872) = 119870 then 119875119877119886119889(119872119870) = 0(3) If119872 = oplus

119894isin119868119872119894is a direct sum of submodules119872

119894 then

PRad (119872) =⨁

119894isin119868

PRad (119872119894) (3)

Proof Let 119864 = 1198901 119890


triangulating idempotents of 119877

(1) Let 119870 be any piecewise prime submodule of 119872 withrespect to 119864 If119873 sube 119870 then PRad(119873) sube 119870 If119873 sube 119870then by the definition it is easy to see that 119873 cap 119870 isa piecewise prime submodule of119873 with respect to 119864Thus PRad(119873) sube (119870 cap 119873) sube 119870 Hence PRad(119873) subePRad(119872)

(2) Let 119875119870 be a piecewise prime submodules of 119872119870

with respect to 119864 By definition (119872119870)(119875119870) is apiecewise primemodule with respect to 119864 Thus119872119875

is a a piecewise prime module with respect to 119864 Thisimplies that 119875 is a piecewise prime submodules of119872with respect to 119864 Hence PRad(119872119870) = 0

(3) By (1) for each 119894 isin 119868 PRad(119872119894) sube PRad(119872) This

implies that

⨁

119894isin119868

PRad (119872119894) sube PRad (119872) (4)

Let (119898119894)119894isin119868

isin 119872 oplus119894isin119868PRad(119872

119894) Then there exists 119894 isin 119868

such that 119898119894notin PRad(119872

119894) By the definition there exists a

piecewise prime submodule119873119894sube 119872119894with respect to 119864 such

that 119898119894notin 119873119894 If 119870 = 119873

119894oplus (oplus119894 = 119895119872119895) then 119870 is a piecewise

4 Algebra

prime submodule of119872 with respect to 119864 and 119898 notin 119870 Thus119898 notin PRad(119872) It means that

PRad (119872) =⨁

119894isin119868

PRad (119872119894) (5)

References

[1] G F Birkenmeier H E Heatherly J Y Kim and J K ParkldquoTriangularmatrix representationsrdquo Journal of Algebra vol 230no 2 pp 558ndash595 2000

[2] J Dauns ldquoPrime modulesrdquo Journal fur die Reine und Ange-wandte Mathematik vol 298 pp 156ndash181 1978

[3] E H Feller and E W Swokowski ldquoPrime modulesrdquo CanadianJournal of Mathematics vol 17 pp 1041ndash1052 1965

[4] M Behboodi O A S Karamzadeh and H Koohy ldquoModuleswhose certain submodules are primerdquo Vietnam Journal ofMathematics vol 32 no 3 pp 303ndash317 2004

[5] M Behboodi and H Koohy ldquoOn minimal prime submodulesrdquoFar East Journal of Mathematical Sciences vol 6 no 1 pp 83ndash88 2002

[6] S T Rizvi and C S Roman ldquoBaer and quasi-Baer modulesrdquoCommunications in Algebra vol 32 no 1 pp 103ndash123 2004

[7] T Y Lam A First Course in Noncommutative Rings SpringerNew York NY USA 1991

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2 Algebra

(2) Let 119873 be a submodule of 119872 Then 119873 is a piecewiseprime submodule of119872 with respect to 119864 if119872119873 is aPWP module with respect to 119864

(3) 119872 is piecewise endoprime (PWEP) with respect to acomplete set of left triangulating idempotents 119865 =

1198871 119887

119898 of 119878 such that for each nonzero submod-

ule 119873 sube 119872 119891 isin 119878 and 119887119894isin 119865 if 119891119887

119894119873 = 0 then

119891119887119894= 0

By Definition 1 119873 is a piecewise prime submodule of119872with respect to a set of left triangulating idempotents 119864 if forany119898 isin 119872 119890

119894isin 119864 and 119887 isin 119877

119898119890119894119877119890119894119887 sube 119873 997904rArr 119898119890

119894isin 119873 or 119872119890

119894119887 sube 119873 (2)

Example 2 Let 119864 = 1198901 119890


triangulating idempotents o 119877

(1) Let 1198961and 119896

2be two fields and 119877 = 119896

1times 1198962 Then

119872 = 1198771198961oplus 119877119896

2is not a prime module but it is

piecewise prime with respect to (1 0) (0 1)(2) If 119872 is a prime 119877-module then it is piecewise

prime with respect to any set of left triangulatingidempotents of 119877

(3) Homomorphic image of 119872119877needs to be PWP with

respect to 119864 For example ZZ is a PWP modulewith respect to 0 1 but Z

4is not PWP because

rann119903(2) = rann

119903(Z4)

Corollary 3 If119872 is a PWP 119877-module with respect to 119864 thenany submodule of119872 is PWP with respect to 119864

Proof It can be seen by Definition 1

Proposition 4 Let 119877 be a ring with finite triangulatingdimension

(1) 119868 is a PWP ideal of 119877 if and only if 119877119868 is a PWP 119877-module

(2) 119877 is a PWP ring if and only if 119877119877is PWP

Proof The part one is obtained by Definition 1 and forsecond let 119868 = 0 in part one

Proposition5 Let119872 be an119877-module and let119864 = 1198901 119890

119899

be a set of left triangulating idempotents of 119877 Then thefollowing statements are equivalent

(1) 119872 is PWP with respect to 119864(2) for each 119873 sube 119872 ideal 119868 in 119877 and 119890

119894isin 119864 if 119873119890

119894119868 = 0

then119873119890119894= 0 or119872119890

119894119868 = 0

(3) for each (119898) sube 119872 ideal (119886) in 119877 and 119890119894isin 119864 if

(119898)119890119894(119886) = 0 then (119898)119890

119894= 0 or119872119890

119894(119886) = 0

Proof (1) rArr (2) If 119873119890119894= 0 then there exists 119899 isin 119873 such

that 119899119890119894= 0 and for any 119887 isin 119868 119899119890

119894119877119890119894119887 = 0 By Definition 1

for each 119887 isin 119868119872119890119894119887 = 0 This implies that119872119890

119894119868 = 0

(2) rArr (3) In (2) let119873 = (119898) and 119868 = (119886)(3) rArr (1) Let 119898119890

119894119877119890119894119887 = 0 where 119898 isin 119872 119890

119894isin 119864 and

119887 isin 119877 Thus 119898119890119894119877119890119894119877119890119894119887119877 = 0 or (119898119890

119894)119890119894(119890119894119887) = 0 By (3)

(119898119890119894)119890119894= 0 or 119872119890

119894(119890119894119887) = 0 This implies that 119898119890

119894= 0 or

119872119890119894119887 = 0

Proposition 6 Let119872 be an 119877-module 119878 = 119864119899119889119877(119872) let 119864 =

1198901 119890

119899 be a complete set of left triangulating idempotents of

119877 and let119865 = 1198871 119887



(1) 119872 is a PWP 119877-module with respect to 119864 if and only iffor each119873 sube 119872with119873119890

119894= 0 119886119899119899

119903(119873119890119894) = 119886119899119899

119903(119872119890119894)

(2) If 119872119877

is PWP 119877-module with respect to 119864 then119886119899119899119903(119872) is a PWP ideal of 119877 with respect to 119864

(3) If119872119877is PWEP with respect to 119865 and retractable then

119886119899119899119903(119872) is a PWP ideal of 119877 with respect to 119864

Proof (1) If 119887 isin ann119903(119873119890119894) then there exists 119899 isin 119873 such that

119899119890119894119877119890119894= 0 and 119899119890

119894119877119890119894119887 = 0 Since119872 is PWP 119877-module with

respect to 119864 by Definition 1 119872119890119894119887 = 0 Hence ann

119903(119873119890119894) =

ann119903(119872119890119894) Conversely let119898119890

119894119877119890119894119887 = 0 where 119890

119894isin 119864119898 isin 119872

119887 isin 119877 and 119898119890119894= 0 Thus 119887 isin ann

119903((119898119890119894119877)119890119894) which means

119887 isin ann119903(119872119890119894) or119872119890

119894119887 = 0

(2) Let 119868119890119894119869 sube ann

119903(119872) and 119868119890

119894sube ann

119903(119872) Since

(119872119868119890119894)119890119894119869 = 0 and119872 is a PWP 119877-module with respect to 119864

by Proposition 5119872119890119894119869 = 0Thus 119890

119894119869 sube ann

119903(119872)This implies

that ann119903(119872) is a PWP ideal of 119877 with respect to 119864

(3) Let 119868119890119894119869 sube ann

119903(119872) where 119868119890

119894 119890119894119869 sube ann

119903(119872) Since

119872 is retractable then there exists a nonzero homomorphism119891 119872 rarr 119872119868119890

119894 There exists 119887

119895isin 119865 such that 119891119887

119895= 0

Since 119868119890119894119869 sube ann

119903(119872) 119891119887

119895119872119890119894119869 = 0 By assumption 119872 is

PWEP with respect to 119865 This implies that 119891119887119895= 0 which

is a contradiction Hence ann119903(119872) is a PWP ideal of 119877 with

respect to 119864

A module 119872119877is called retractable if for any nonzero

submodule119873 of119872 Hom119877(119872119873) = 0

Theorem 7 Let 119872 be an 119877-module 119878 = 119864119899119889119877(119872) and

let 119865 = 1198871 119887



(1) If119878119872 is a PWP module with respect to 119865 then 119878 is a

PWP ringThe converse is true when119872119877is retractable

(2)119878119872 is a PWP module with respect to 119865 if and only if119872119877is PWEP with respect to 119865

Proof (1) Let 119891119887119894119878119887119894119892 = 0 where 119891 119892 isin 119878 119887

119894isin 119865 and 119887

119894119892 = 0

Thus there exists119898 isin 119872 such that 119887119894119892119898 = 0 and 119891119887

119894119878119887119894119892119898 =

0 Since119878119872 is PWPwith respect to 119865119891119887

119894119872 = 0whichmeans

119891119887119894= 0 Conversely let 119891119887

119894119878119887119894119898 = 0 and 119887

119894119898 = 0 Since 119872

119877

is retractable there exists a nonzero homomorphism 119887119894119892 isin

Hom119877(119872 119887119894119898119877) Thus 119891119887

119894119878119887119894119892 = 0 Since 119878 is PWP 119891119887

119894= 0

(2) Assume119872 is a PWP 119878-module with respect to 119865 Let119873 sube 119872 and 119891119887

119894119873 = 0 where 119891 isin 119878 and 119887

119894isin 119865 Since

119878119872 is PWP by Proposition 6(1) 119891119887

119894119872 = 0 Thus 119891119887

119894=

0 Conversely assume 119872119877be PWEP with respect to 119865 Let

119891119887119894119878119887119894119898 = 0 where 119891 isin 119878 119887

119894isin 119865 119898 isin 119872 and 119887

119894119898 = 0

If 119873 = 119878119887119894119898 then 119891119887

119894119873 = 0 This implies that 119891119887

119894= 0 or

119891119887119894119872 = 0 Hence

119878119872 is PWP with respect to 119865

Algebra 3


119877


119878(119873) =

119878119890 where 119890 = 1198902 isin 119878 [6]


and let 119865 = 1198871 119887





(3) 119872119877is quasi-Baer





119899





(2) Let119872 = prod120582isinΛ




120582isin 119872

120582 119890119894isin 119864 and 119887 isin

119877 then (0 119898120582 0 0)119890

119894119877119890119894119887 = 0 Since 119872 is PWP

(0 119898120582 0 0)119890

119894= 0 or 119872119890

119894119887 = 0 This implies

that 119898120582119890119894= 0 or 119872




(1198981 119898

119899 0 )119890


120582119890119894119877119890119894119887 = 0


120582119890119894= 0 or119872119890

119894119887 = 0

Hence (1198981 119898

119899 0 )119890

119894= 0 or119872119890

119894119887 = 0 Thus119872 is PWP


Corollary 10 Let 119864 = 1198901 119890









119877is quasi-

Baer


119872 lann119878(119873) = lann


119877is quasi-


lann119878(119890119872) = lann

119878((1minus119890)119872) = lann


119890 = 1 or 119890 = 0 Thus 120591 dim(119878) = 1


119877is quasi-


119903(119878) Since 120591 dim(119878) = 1 119890 isin


119878119872 lann

119878(119873) = lann

119878(119872) =




1 119890







(2) If 119875119877119886119889(119872) = 119870 then 119875119877119886119889(119872119870) = 0(3) If119872 = oplus


119894 then

PRad (119872) =⨁

119894isin119868

PRad (119872119894) (3)

Proof Let 119864 = 1198901 119890








implies that

⨁

119894isin119868

PRad (119872119894) sube PRad (119872) (4)

Let (119898119894)119894isin119868






that 119898119894notin 119873119894 If 119870 = 119873


4 Algebra


PRad (119872) =⨁

119894isin119868

PRad (119872119894) (5)

References










Game Theory




Complex Systems

Journal of















ISRN Combinatorics



Geometry






DecisionSciences





Algebra





Volume 2013








Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Algebra 3


119877


119878(119873) =

119878119890 where 119890 = 1198902 isin 119878 [6]


and let 119865 = 1198871 119887





(3) 119872119877is quasi-Baer





119899





(2) Let119872 = prod120582isinΛ




120582isin 119872

120582 119890119894isin 119864 and 119887 isin

119877 then (0 119898120582 0 0)119890

119894119877119890119894119887 = 0 Since 119872 is PWP

(0 119898120582 0 0)119890

119894= 0 or 119872119890

119894119887 = 0 This implies

that 119898120582119890119894= 0 or 119872




(1198981 119898

119899 0 )119890


120582119890119894119877119890119894119887 = 0


120582119890119894= 0 or119872119890

119894119887 = 0

Hence (1198981 119898

119899 0 )119890

119894= 0 or119872119890

119894119887 = 0 Thus119872 is PWP


Corollary 10 Let 119864 = 1198901 119890









119877is quasi-

Baer


119872 lann119878(119873) = lann


119877is quasi-


lann119878(119890119872) = lann

119878((1minus119890)119872) = lann


119890 = 1 or 119890 = 0 Thus 120591 dim(119878) = 1


119877is quasi-


119903(119878) Since 120591 dim(119878) = 1 119890 isin


119878119872 lann

119878(119873) = lann

119878(119872) =




1 119890







(2) If 119875119877119886119889(119872) = 119870 then 119875119877119886119889(119872119870) = 0(3) If119872 = oplus


119894 then

PRad (119872) =⨁

119894isin119868

PRad (119872119894) (3)

Proof Let 119864 = 1198901 119890








implies that

⨁

119894isin119868

PRad (119872119894) sube PRad (119872) (4)

Let (119898119894)119894isin119868






that 119898119894notin 119873119894 If 119870 = 119873


4 Algebra


PRad (119872) =⨁

119894isin119868

PRad (119872119894) (5)

References










Game Theory




Complex Systems

Journal of















ISRN Combinatorics



Geometry






DecisionSciences





Algebra





Volume 2013








Volume 2014




Journal of











Journal of


Function Spaces






Algebra









4 Algebra


PRad (119872) =⨁

119894isin119868

PRad (119872119894) (5)

References










Game Theory




Complex Systems

Journal of















ISRN Combinatorics



Geometry






DecisionSciences





Algebra





Volume 2013








Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Game Theory




Complex Systems

Journal of















ISRN Combinatorics



Geometry






DecisionSciences





Algebra





Volume 2013








Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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