3 2012 333342 ISSN 13118080 printed version url httpwwwijpameu ijpameu ON COFINITELY GENERALIZED AMPLY WEAK SUPPLEMENTED MODULES Figen Y57512uzba57528si S57528enol Eren Department of Mathematics Faculty of Sciences and Arts Ondokuz May305s Universi ID: 35522 Download Pdf

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3 2012 333342 ISSN 13118080 printed version url httpwwwijpameu ijpameu ON COFINITELY GENERALIZED AMPLY WEAK SUPPLEMENTED MODULES Figen Y57512uzba57528si S57528enol Eren Department of Mathematics Faculty of Sciences and Arts Ondokuz May305s Universi

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International Journal of Pure and Applied Mathematics Volume 76 No. 3 2012, 333-342 ISSN: 1311-8080 (printed version) url: http://www.ijpam.eu ijpam.eu ON (COFINITELY) GENERALIZED AMPLY WEAK SUPPLEMENTED MODULES Figen Yuzbasi , Senol Eren Department of Mathematics Faculty of Sciences and Arts Ondokuz Mayıs University 55139, Kurupelit-Samsun, TURKEY Abstract: Let be a ring and be a left module. In this paper, we will study someproperties of (coﬁnitely) generalized ampl y weak supplemented modules (CGAWS) as a generalization of

(coﬁnitely) amply we ak supplemented andgive anewcharacterization of semilocal ringsusingCGA WS-modules. Nev- ertheless, we will show that (1) is Artinian if and only if is a GAWS- module and satisﬁes DCC on generalized weak supplement subm odules and on small submodules. (2) A ring is semilocal if and only if every left -module is CGAWS-module. AMS Subject Classiﬁcation: 16D10, 16D99, 16P70 Key Words: coﬁnitely generalized weak supplemented module, generali zed amply weak supplemented module, coﬁnitely generalized amp ly weak supple- mented module 1.

Introduction and Preliminiaries Throughout the paper, will be an associative ring with identity and all modules are unital left modules unless otherwise speciﬁed. Let be an module. The symbol means that is a submodule of . Recall that asubmodule is called small anddenoted by if Received: October 20, 2011 2012 Academic Publications, Ltd. url: www.acadpubl.eu Correspondence author

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334 F. Yuzbasi, S. Eren for every proper submodule of Rad ) will indicate Jacobson radical of is a supplement of in if and only if and (see [7]) where and are submodules

of is called supplemented , if every submodule of has a supplement in , i.e. a submodule minimal with respect to .If and , then is called a weak supplement of in , (see [5], [9]), and clearly in this situation is a weak supplement of , too. is a weakly supplemented module if every submodule of has a weak supplement in A submodule of a is said to be coﬁnite if is ﬁnitely generated. is called a coﬁnitely (weak) supplemented module if every coﬁnite submodule of has a (weak) supplement in (see [1], [2]). Clearly supplemented modules are coﬁnitely supplemented and

weakly supplemented module s are coﬁnitely weak supplemented. A submodule of a module has ample (weak) supplements in if for all with , there is a (weak) supplement of with . If every submodule of has ample (weak) supplements in , then is called amply (weak) supplemented . Similarly, if every coﬁnite submodule of has ample (weak) supplements in is called coﬁnitely amply (weak) supplemented Let be an module and be any submodules of with K. If Rad ) ( Rad )) then is called a gener- alized (weak) supplement of in . Following [6], is called generalized supplemented module or

brieﬂy a GS -module if every submodule of has a generalized supplemented in . In [6], an module is called gener- alized weakly supplemented or brieﬂy a GWS -module ( WGS -module in [6]) if every submodule of has a generalized weak supplement in is called a generalized amply supplemented module or brieﬂy a GAS -module in case implies that has a generalized supplement . For characterizations of generalized (amply) supplemented an d generalized weakly supplemented modules we refer to [6] and [8]. is called coﬁnitely generalized supplemented if every coﬁnite

submodule of has a generalized supplement [4]. In this paper, we introduce generalized amply weak suppleme nted modules and coﬁnitely generalized (amply) weak supplemented modul es. We obtain some properties of these modules and have the following impl ications of these modules:

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ON (COFINITELY) GENERALIZED AMPLY WEAK... 335 Generalized Amply Supplemented Generalized Amply Weak Supplemented Generalized Weak Supplemented Cofinitely Generalized Weak Supplemented 2. Coﬁnitely Generalized Weak Supplemented Modules Deﬁnition 1. A module is called a

coﬁnitely generalized weak sup- plemented or brieﬂy a CGWS-module if every coﬁnite submodul e of has a generalized weak supplement. To prove that an arbitrary sum of CGWS-modules is a CGWS-modu le, we use the following standard lemma. Lemma 2. Let be a module, and be submodules of with coﬁnitely generalized weak suplemented and coﬁnite . If has a generalized weak supplement in , then also has a generalized weak supple- ment in Proof. Let be a generalized weak supplement of in . Then we have )] +( Since is a coﬁnite submodule, is a ﬁnitely

generated module. The last module in the right hand side of the preceding equation is a ﬁn itely generated module hence ) has a generalized weak supplement in , i.e. +[ )] = )] = Rad Rad Since +[ )] = Y,

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336 F. Yuzbasi, S. Eren is a generalized weak supplement of in . Therefore )]+[ )] Rad This means that is a generalized weak supplement of in Proposition 3. Any arbitrary sum of CGWS-modules is a CGWS- module. Proof. Let where each module is a coﬁnitely generalized weak supplemented and bea coﬁnite submoduleof . Then is generated

bysomeﬁniteset N,x N,....,x andtherefore Rx Rx ... Rx . Since each is contained in the sum for some ﬁnite subset ,..., (1) ,..,n of −{ has a trivial generalized weak supplement 0 in and since is a CGWS-module, has a generalized weak supplement by Lemma 2. Continuing in this way we will obtain (after we have used Lemma 2 =1 ) times) at last has a generalized weak supplement in 3. Generalized Amply Weak Supplemented Modules In this section, we deﬁne the concept of generalized amply we ak supplemented modules, which is adapted from amply weak supplemented modu les, and

we give the properties of these modules. Deﬁnition 4. Let be a module and . If every with there exists a generalized weak supplement of with then we call has generalized ample weak supplements in Deﬁnition 5. Let be a module. If every submodule of has a generalized ampleweak supplementsin M, then is called ageneralized amply weak supplemented module or brieﬂy GAWS-module. Proposition 6. Any factor module of a GAWS-module is a GAWS- module. Proof. Let be a GAWS-module, be any submodule of and be any submodule of . For , let . Then and

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ON (COFINITELY)

GENERALIZED AMPLY WEAK... 337 there exists a generalized weak supplement of with since is a GAWS-module. Therefore Now, let be a canonical epimorphism. Since Rad ), we obtain (( )+ Rad )) Rad Also implies that has a generalized weak supplement in Therefore is a GAWS-module. Corollary 7. Any homomorphic image of a GAWS-module is a GAWS- module. Lemma 8. Every supplement submodule of a GAWS-module is general- ized amply weak supplemented. Proof. Let be a GAWS-module and be any supplement submodule of . Suppose that is a supplement of in . Let and for . Then . Since is generalized amply weak

supplemented, has a generalized weak supplement in with In this case and ( Rad Since and is a supplement of in , one can see that . Therefore Rad Rad ). (See [7], 41.1) for Rad Rad ) and is generalized amply weak supplemented. Corollary 9. Every direct summand of a GAWS-module is generalized amply weak supplemented. Theorem 10. Let be a module and . If and have generalized ample weak supplements in , then has also generalized ample weak supplements in Proof. Let and = ( )+ . Then we have = ( )+( = ( )+(

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338 F. Yuzbasi, S. Eren and so and Since and have

generalized ample weak supplements in , there exist and such that and Rad and and Rad Therefore and = ( )+ and = ( )+ . As a result, we get = ( )+( and +( = ( )+( Rad which completes the proof. A module is said to be projective if for any two submodules and of with there exists End ) with and (1 )( Proposition 11. Let be a module. If is a projective GWS- module, then is a GAWS-module. Proof. Let and be two submodules of , such that . Since is projective, there exists an endomorphism of such that and (1 )( . Let be a generalized weak supplement of in Then we have )+(1 )( ) = )+(1 )( +(1 )( and

so + (1 )( ). It is easy to see that (1 )( . Let (1 )( ). Then and = (1 )( ) = ), for some . Being implies that (1 )( ). However Rad ) gives that (1 )( ) = (1 )( Rad ((1 )( )) Thus (1 )( ) is a generalized weak supplement of in and is a GAWS-module.

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ON (COFINITELY) GENERALIZED AMPLY WEAK... 339 Corollary 12. Every projective and GWS-module is generalized amply weak supplemented. Corollary 13. Let ,M ,....M be projective modules. Then =1 generalized amply weak supplemented if and only if is a generalized amply weak supplemented for every Proof. The necessity is obvious by

Corollary 9. Since for every 1 is a generalized amply weak supplemented, is a generalized weak sup- plemented. Therefore =1 is also generalized weak supplemented by ([6], Proposition 3.7). Since for every 1 is projective, =1 is also projective. Then by Corollary 12, =1 generalized amply weak supple- mented. Theorem 14. Let be a module. Then is Artinian if and only if is a GAWS-module and satisﬁes DCC on generalized weak supple ment submodules and on small submodules. Proof. The necessity is clear. Conversely, suppose that is a GAWS- module which satisﬁes DCC on generalized weak

supplement su bmodules and on small submodules. Then Rad ) is Artinian by ([3], Theorem 5). Next it suﬃces to show that Rad is Artinian. Let be any submodule of containing Rad ). Then there exists a generalized weak supplement of in , i.e. and Rad ). Therefore, we have Rad Rad Rad Rad and that every submodule of Rad is a direct summand. It means Rad is semisimple. Now suppose that Rad ..... is an ascending chain of submodules of . Because is a GAWS-module, there exists a descending chain of submodules .... such that is a generalized weak supple- ment of in for each 1. By hypothesis,

there exists a positive integer such that +1 +2 .... . Because of Rad Rad Rad )) Rad for all , it follows that +1 +2 .... . Thus Rad is Noetherian and ﬁnitely generated. This means Rad is Artinian by ([7], 31.3).

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340 F. Yuzbasi, S. Eren 4. Coﬁnitely Generalized Amply Weak Supplemented Modules In this section we deﬁne and study coﬁnitely generalized amp ly weak supple- mented modules. Deﬁnition 15. An module is called coﬁnitely generalized amply weak supplemented, or brieﬂy CGAWS-module if every

coﬁnite submodule of has a generalized ample weak supplement in Proposition 16. Let be a CGAWS-module. Then (1) Every supplement submodule of is a CGAWS-module. (2) Every factor module of is a CGAWS-module. Proof. (1) Let be a supplement of in and is a coﬁnite submodule of . Then we have )+ Since the last module in the right hand-side of the preceding equation is a ﬁnitely generated module, we get that ( )+ is also a coﬁnite submodule of . Let for any . Then . Since is a coﬁnite submodule of and is a CGAWS-module, has a generalized weak supplement in with ,

i.e. M, Rad Also we have because of and is a supplement of in . Thus Rad Rad ). Hence is a CGAWS-module. (2) Let be a submodule of and is a coﬁnite submodule of . Note that . Hence is a coﬁnite submodule of . For , let . This implies that . Since is a CGAWS-module, there exists a generalized weak supplement of with . Therefore . Let be a canonical epimorphism. Since Rad ), we have (( )+

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ON (COFINITELY) GENERALIZED AMPLY WEAK... 341 Rad )) Rad If one uses , then he can see that has a generalized ample weak supplement in which completes the proof. Corollary 17.

(1)EverydirectsummandofaCGAWS-moduleiscoﬁnitely generalized amply weak supplemented. (2)Every homomorphic image of a CGAWS-module is a CGAWS-mod ule. Proposition 18. Let be a coﬁnitely generalized weak supplemented and projective module. Then is coﬁnitely generalized amply weak sup- plemented. Proof. Let be a coﬁnite submodule of and for Suppose that be a generalized weak supplement of in . Since is projective, there exists a homomorphism such that and (1 )( . In this case, we get )+(1 )( )+ and so ). Let ). Then there exists with ). If we write ) = (1 )( then we

get and . Hence and Rad )) = Rad )) Rad As a result, ) is a generalized weak supplement of in and so is a CGAWS-module. Corollary 19. Every projective and CGWS-module is coﬁnitely general- ized amply weak supplemented. Corollary 20. Let be a family of projective modules. Then is coﬁnitely generalized amply weak supplemented if and onl y if for every is coﬁnitely generalized amply weak supplemented. Proof. The necessity is obvious by Corollary 17(1). Since for every is coﬁnitely generalized amply weak supplemented, is coﬁnitely gener- alized weak supplemented

for all I. Then by Proposition 3, is also coﬁnitely generalized weak supplemented. Since for every is pro- jective, is also projective. Therefore is coﬁnitely generalized amply weak supplemented by Corollary 19.

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342 F. Yuzbasi, S. Eren Corollary 21. Let be a ring. Then the following statements are equiv- alent: (i) is semilocal. (ii) Every left module is generalized weakly supplemented. (iii) Every left module is generalized amply weak supplemented. (iv) Every left module is coﬁnitely generalized weak supplemented. (v) Every left

module is coﬁnitely generalized amply weak supplemented. Proof. The implications ( ii ) and ( ii ) can be seen in [5]. The others, ( ii iii ii iv ) and ( ii ) are obvious. References [1] R. Alizade, G. Bilhan, P.F.Smith, Modules whose maximal submodules have supplements, Communications in Algebra 29 (2001), 2389-2405. [2] R. Alizade, E. Buyukasık, Coﬁnitely weakly suppleme nted modules, Com- mutative Algebra 31 (2003), 5377-5390. [3] I. Al-Khazzi, P.F. Smith, Modules with chain conditions on superﬂuous submodules, Comm. Algebra 19 (1991),

2331-2351. [4] E. Buyukasık, C. Lomp, On a recent generalization of s emiperfect rings, Bulletin of the Australian Mathematicial Society 78 (2008), 317-325. [5] C. Lomp, On semilocal modules and rings, Communications in Algebra 27 (1999), 1921-1935. [6] Y. Wang, N. Ding, Generalized supplemented modules, Taiwanese Journal of Mathematics 10 (2006), 1589-1601. [7] R. Wisbauer, Foundations of Module and Ring Theory , Gordon and Breach, Philadelphia (1991). [8] W. Xue, Characterizations of semiperfect and perfect ri ngs, Publications Matematiques 40 (1996), 115-125.

[9] H. Zoschinger, Invarianten wesentlicher uberdeckun gen, Math. Annalen 237 (1978), 193-202.

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