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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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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Download Pdf - The PPT/PDF document "International Journal of Pure and Applie..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.

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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 Zoschinger’s works ([9], [10], [11], [12]). After Zoschinger, 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 , Birkhauser 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. Yuzbası 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. Zoschinger, Komplementierte Moduln uber Dedekind ringen, Journal of Algebra, 29 (1974), 42-56. [10] H.Zoschinger, KomplementealsdirekteSummanden, Arch. Math. (Basel), 25 (1974), 241-253. [11] H.

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

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