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4 2013 613621 ISSN 13118080 printed version ISSN 13143395 onli ne version url httpwwwijpameu doi httpdxdoiorg1012732ijpamv83i410 ijpameu AMPLY WEAK SEMISIMPLESUPPLEMENTED MODULES Figen Takil Mutlu Department of Mathematics Anadolu University Eskiseh ID: 35521 Download Pdf

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International Journal of Pure and Applied Mathematics Volume No

4 2013 613621 ISSN 13118080 printed version ISSN 13143395 onli ne version url httpwwwijpameu doi httpdxdoiorg1012732ijpamv83i410 ijpameu AMPLY WEAK SEMISIMPLESUPPLEMENTED MODULES Figen Takil Mutlu Department of Mathematics Anadolu University Eskiseh

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International Journal of Pure and Applied Mathematics Volume 83 No. 4 2013, 613-621 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-li ne version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v83i4.10 ijpam.eu AMPLY WEAK SEMISIMPLE-SUPPLEMENTED MODULES Figen Takil Mutlu Department of Mathematics Anadolu University Eskisehir, 26470, TURKEY Abstract: Let be a ring and be a right -module. In this paper we will study various properties of amply weak semisimple-supplem ented module. It is shown that: (1) every projective weakly semisimple-supp lemented

module is amply weak semisimple-supplemented; (2) if is an amply weak semisimple- supplemented module and satisfies DCC on weak semisimple-su pplement sub- modules and on small submodules, then is Artinian; (3) an amply weak semisimple-supplemented modulebehaves well with respect to supplements and to homomorphic images. AMS Subject Classification: 16D10, 16D60, 16D70, 16D99 Key Words: supplement submodule, weak semisimple-supplement submod ule, amply weak semisimple-supplemented module 1. Introduction Throughout this article, all rings are associative with uni ty and denotes such

a ring. All modules are unital right -modules unless indicated otherwise. Let be an -module. will mean is a submodule of Soc ), End ) and Rad ) will denote the Socle of , the ring of endomorphisms of M and the Jacobson radical of , respectively. The notions which are not explained here will be found in [7]. Received: January 7, 2013 2013 Academic Publications, Ltd. url: www.acadpubl.eu
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614 F.T. Mutlu Recall that a module is called semisimple if it is a direct sum of simple submodules. A submodule is called small in (notation K << M ) if for every submodule in , the equality

implies . A module is called hollow if every proper submodule of is small (see, [7]). Let and be submodules of is called a supplement of in if is a minimal element in the set of submodules with (see,[3]). In ([4], Definition 4.4, p.56) is called supplemented if any submodule of has a supplement in In early years, supplemented modules and the other generali zation, amply supplemented modules appeared in Helmut Zoschingerís works ([9], [10], [11], [12]). After Zoschinger, many authors (see for example [2] , [5], [6] and [8]) studied on variations of supplemented modules.

This paper i s based on another variation of supplemented modules. We say that a submodule of has ample weak semisimple-supplements in if, for every with , there exists a weak semisimple-supplement of with . We say that is amply weak semisimple-supplemented module if every submodule of has ample weak semisimple-supplements in . We proved that every projective weak semisimple-supplemented module is amply w eak semisimple- supplemented. Itisshownthatif isanamplyweaksemisimple-supplemented module and satisfies DCC on semisimple-supplement submodul es and on small submodules, then is

Artinian. Moreover, it is proven that an amply weak semisimple-supplemented modulebehaves well with respect to supplements and to homomorphic images. In this section, we discuss the concept of semisimple-suppl ement submodu- les and we give some properties of such type submodules. Definition 1. Let be an -module, and be two submodules of is called semisimple-supplement of in if S << S and Soc ) = Since is semisimple, every submodule of is a direct summand. If N << S , then = 0. Hence, being a semisimple-supplement of , we have is semisimple and is the minimal element in the set of

submodules with Definition 2. Let be an -module. We say that is semisimple- supplemented if all submodules of has a semisimple-supplement in Definition 3. Let be an -module and . If, for every with , there exists a semisimple-supplement of with then we say that N has ample semisimple-supplements in Definition 4. Let be an -module. If every submodule of has
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AMPLY WEAK SEMISIMPLE-SUPPLEMENTED MODULES 615 ample semisimple-supplements in , then is called amply semisimple-supp- lemented module. It is clear that every amply semisimple-supplemented modul e is amply

sup- plemented. Proposition 5. Let be an -module. Then the following statements are equivalent. (a) is semisimple. (b) is semisimple-supplemented. (c) is amply semisimple-supplemented. Proof. (a)= (b). It is clear. (b)= (c). Let . Since is semisimple-supplemented, there exists a semisimple supplement of in . Then . Hence = ( ) = ). By the minimality of and hence . Thus has ample semisimple supplement with (c)= (a). Let . Since is amply semisimple-supplemented module, there exists a semisimple supplement of in . Then and N << S . Since is semisimple, every submodule of is a direct summand. So =

0 and hence . Thus is semisimple. Definition 6. Let be an -module, N S be two submodules of is called weak semisimple-supplement of in if S << M and Soc ) = Definition 7. Let be an -module. We say that a submodule is a weak semisimple-supplement if it is a weak semisimple-supplement for some submodule Definition 8. Let be an -module. If every nonzero submodule of hasaweaksemisimple-supplementsin , then iscalled a weakly semisimple- supplemented module or briefly a WSS-module. It is clear that every semisimple-supplemented module is we akly semisimple supplemented.

Proposition 9. Let be an -module, be a submodule of where be a weak semisimple-supplement of in . Then the following statements are hold. (1) If for some , then is also a weak semisimple- supplement of in M.
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616 F.T. Mutlu (2) If is finitely generated, then is also finitely generated. (3) If K << M , then is a weak semisimple-supplement of in M. (4) For /K is a weak semisimple-supplement of N/K in M/K Proof. (1) By the definition of weak semisimple-supplement, S << M and is semisimple. If for some , then S << M . Therefore is a weak semisimple-supplement of

in (2) From ([7], 41.1(2)). (3) Let and . Since K << M and S << M . By the minimality of . Then is a weak semisimple-supplement of in (4) By the definition of weak semisimple-supplement, N << and is semisimple. Hence . Therefore M/K N/K +[( /K ]. Now, we show that ( N/K [( /K << M/K Let [( N/K [( /K ]]+ T/K M/K and . Then [ )]+ andbymodularlaw +( )+ . Since S << M and . Hence ( N/K [( /K << M/K Thus ( /K is a supplement of N/K . Finally, since is semisimple, /K is semisimple submodule of M/K Lemma 10. Let be an -module and ,M ,...,M be submodules of . Then is WSS-module if and only if

every (1 is WSS-module. Proof. Let . To prove WSS-module it is sufficient by induction on to prove this when = 2. Thus suppose = 2. Assume that is WSS-module. Let . By assumption has a weak semisimple-supplement in and has a weak semisimple-supplement in . Then << M and << M . Hence = ( ) = ( )+( and ))+( )) << M
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AMPLY WEAK SEMISIMPLE-SUPPLEMENTED MODULES 617 Since and are semisimple, is semisimple. Hence is weak semisimple-supplement of . Thus is WSS-module. Conversely, assume that and are WSS-module. Let . By assumption has a weak semisimple-supplement in . Then ( )+

and ( S << M . Hence (( )+ ) = +( and S << M. Hence << M . Note that is semisimple since it is a submodule of semisimple submodule . Thus is a weak semisimple- supplement of in 2. Amply Weak Semisimple-Supplemented Modules In this section, we discuss the concept of amply weak semisim ple-supplemented modules and we give some properties of such type modules. Definition 11. Let be an -module and . If, for every with , there exists a weak semisimple-supplement of with , then we say that N has ample weak semisimple-supplements in Definition 12. Let be an -module. If every submodule of

has ample weak semisimple-supplements in , then is called an amply weak semisimple-supplemented module or briefly an AWSS-module. Proposition 13. Every AWSS-module is WSS-module. Proof. Let be an AWSS-module and be a submodule of . Then . Since is AWSS-module, contains a weak semisimple- supplement of . Hence is WSS-module. Proposition 14. Let be an -module. If every submodule of is a WSS-module, then is AWSS-module. Proof. Let L,N and . By assumption, there is a weak semisimple-supplement submodule of in . Then ( )+ and S << L . Thus S << M and +( ) = and hence . Therefore , as

required.
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618 F.T. Mutlu Proposition 15. Every factor module of an AWSS-module is AWSS- module. Proof. Let be an AWSS-module and M/K be any factor module of Let N/K M/K . For L/K M/K , let N/K L/K M/K . Then . Since is AWSS-module, there exists a weak semisimple-supplement of with . By Proposition 9(4), ( /K is a weak semisimple- supplement of N/K in M/K . Since ( /K L/K N/K has ample weak semisimple-supplements in M/K . Thus M/K is AWSS-module. Corollary 16. Every homomorphic image of an AWSS-module is AWSS- module. Proof. Let be an AWSS-module. Since every homomorphic image

of is isomorphic to a factor module of , every homomorphic image of is AWSS-module by Proposition 15. Proposition 17. Every supplement submodule of an AWSS-module is AWSS-module. Proof. Let be an AWSS-module and be any supplement submodule of . Then there exists a submodule of such that is a supplement of . Let and for . Then . Since is AWSS-module, hasaweak semisimple-supplement 00 in with 00 In this case ( )+ 00 . Since 00 and is a supplement of in 00 . On the other hand, since 00 00 << M 00 << M . Hence has ample weak semisimple-supplements in . Thus is AWSS-module. Corollary 18. Every

direct summand of an AWSS-module is AWSS- module. Proof. Let be an AWSS-module. Since every direct summand of is supplement in , then by Proposition 17, every direct summand of is AWSS-module. A module is said to be -projective if, for every two submodules N,L of with , there exists End ) with Imf and Im (1 , see [7]. Theorem 19. Let be a WSS-module and -projective module. Then is AWSS-module.
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AMPLY WEAK SEMISIMPLE-SUPPLEMENTED MODULES 619 Proof. Let and for . Since is WSS-module, there exists a weak semisimple-supplement of in . Then S << M and is semisimple. Since is

-projective, there exists an endomorphism such that and (1 )( . Note that and (1 )( . Then )+(1 )( )+ Let ). Thenthereexists with ). Inthiscase ) = (1 )( and then . Hence and ). Since S << M , then by Lemma ([7], 19.3(4)) << f ). Then << M . Since is semisimple, ) is semisimple. Hence ) is a weak semisimple-supplement of in . Since has ample weak semisimple-supplements in Thus is AWSS-module. Corollary 20. Every projective and WSS-module is AWSS-module. Proof. Since every projective module is -projective, every projective and WSS-module is AWSS-module by theorem 19. Corollary 21. Let ,M ,M be

projective modules. Then =1 is AWSS-module if and only if for every is AWSS-module. Proof. (= ) It is clear from Corollary 18. =) Since, for every 1 is AWSS-module, is WSS-module. Then =1 is also WSS-moduleby Lemma10. Since, for every 1 is projective, =1 is also projective. Then =1 is AWSS-module by Corollary 20. Corollary 22. Let be a ring. Then the following statements are equi- valent. (a) is weakly semisimple-supplemented. (b) is amply weak semisimple-supplemented. (c) Every finitely generated -module is AWSS-module. Proof. ). Clear from Corollary 20. ). Clear from Corollary 16 and

Corollary 21.
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620 F.T. Mutlu Theorem 23. ([1],Theorem 5) Let be any ring and be a module. Then Rad is Artinian if and only if satisfies DCC on small submodules. Proposition 24. Let be an -module. If is an AWSS-module and satisfies DCC on weak semisimple-supplement submodules and on small submodules then is Artinian. Proof. Let beanAWSS-modulewhichsatisfiesDCConweaksemisimple- supplement submodules and on small submodules. Then Rad ) is Artinian by Theorem 23. It suffices to show that M/Rad ) is Artinian. Let be any submodule of containing Rad ). Then

there exists a weak semisimple- supplement of in . Hence S << M . Since Rad ), M/Rad ) = ( N/Rad )) (( Rad )) /Rad )) and so ev- ery submodule of M/Rad ) is a direct summand. Therefore M/Rad ) is semisimple. Now suppose that Rad is an ascending chain of submodules of . Because is AWSS-module, there exists a descending chain of submodules such that is a weak semisimple- supplementof in foreach 1. Byhypothesis,thereexistsapositiveinte- ger such that +1 +2 . Because M/Rad ) = /Rad Rad )) /Rad ) for all , it follows that +1 . Thus M/Rad ) is Noetherian, and hence finitely generated. So M/Rad

) is Artinian, as desired. Corollary 25. Let beafinitely generated AWSS-module. If satisfies DCC on small submodules, then is Artinian. Proof. Since M/Rad )issemisimpleand isfinitelygenerated, M/Rad is Artinian. Now that satisfies DCC on small submodules, Rad ) is Ar- tinian by Theorem 23. Thus is Artinian. References [1] I. Al-Khazzi, P.F. Smith, Modules with chain conditions on superfluous submodules, Comm. Algebra 19 (1991), 2331-2351. [2] G.F. Birkenmeier, F. Takıl Mutlu, C. Nebiyev, N. Sokmez, A. Tercan, Goldie*- supplemented Modules, Glasgow Math. J.

52A (2010), 41-52. [3] J. Clark, C. Lomp, N. Vanaja, R. Wisbauer, Lifting Modules: Supplements and Projectivity in Module Theory , Birkhauser Verlag, Basel, Switzerland (2006).
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AMPLY WEAK SEMISIMPLE-SUPPLEMENTED MODULES 621 [4] S.H. Mohamed, B.J. Muller, Continuous and Discrete Modules, London Mathematical Society Lecture Note Series, 147 , Cambridge University Press, Cambridge (1990). [5] Y. Talebi, A. Mahmoudi, On Rad- -supplemented modules, Thai Journal of Mathematics , No. 2 (2011), 373-381. [6] Y. Wang, N. Ding, Generalized supplemented modules, Taiwanese Journal

of Mathematics, 10 , No. 6 (2006), 1589-1601. [7] R. Wisbauer, Foundations of Module and Ring Theory , Gordon and Breach, Philadelphia (1991). [8] F. Yuzbası Eryılmaz, S. Eren, On (cofinitely) general ized amply weak sup- plemented modules, International Journal of Pure and Applied Mathemat- ics 76 , No. 3 (2012), 333-342. [9] H. Zoschinger, Komplementierte Moduln uber Dedekind ringen, Journal of Algebra, 29 (1974), 42-56. [10] H.Zoschinger, KomplementealsdirekteSummanden, Arch. Math. (Basel), 25 (1974), 241-253. [11] H.

Zoschinger, Komplemente als direkte Summanden II, Arch. Math. (Basel), 38 , No. 4 (1982), 324-334. [12] H. Zoschinger, Komplemente als direkte Summanden III Arch. Math. (Basel), 46 , No. 2 (1986), 125-135.
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