5008

4712

كارشناسي ارشد

رياضي محض﴿جبر﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1388

[هفت] ،77ص.

ص.ع. به: فارسي و انگليسي

محمد رضا ودادي

عاطفه قرباني

19/11/88

منصور معتمدي، احمد حقاني

رياضي

ID4712

به فارسي و انگليسي: قابل رويت در نسخه ديجيتال

Generalized Supplemented Modules Zeinab Kushki z koushki@math iut ac ir November 02 2009 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Mohammad Reza Vedadi mrvedadi@cc iut ac ir2000 MSC Primary 16P 70 Secondary 16D10Key words Generalized supplemented module Artinian module Liftingmodule Semilocal ring Abstract In this thesis we present an expanded account of Generalized supplemented Modulesbased on an article by Yongduo Wang and Nanqing Ding 2006 we assume that R is a ring not necessarily commutative associative with identity and Mis an unital right R modules unless otherwise speci ed The concepts of generalized amply supplemented modules were introduced in to characterize semiperfect modules semi perfectrings A module M is called GS module If for any submodule U of M there exists a submodule Uof M such that M U U and U U Rad U It is well known that a module M is Artinian if and only if M is an amply supplementedmodule and satis es DCC on supplement submodules and on small submodules A module M is called a generalized amply supplemented module or brie y a GAS module incase M U V implies that U has a generalized supplement U V In chapter 4 we show that a module M is Artinian if and only if M is a GAS module andsatis es DCC on generalized supplement submodules and on small submodules It is also proven that a module M with ACC on small submodules is a lifting module if andonly if M is a GAS module and every generalized supplement is a direct summand of M ifand only if M satis es P In chapter 5 we de ne the concept of a weakly generalized supplemented module or brie ya WGS module A module M is said to be a generalized weakly supplemented or brie y a WGS module if forany submodule N M there exists L M such that M N L and N L Rad M LetM be a module and Rad M M Then M is a WGS module if and only if M Rad M issemisimple if and only if There is a decomposition M M1 M2 such that M1 is semisimple Rad M e M2 and M2 Rad M is semisimple also we prove that for module M the following statements are equivalent 1 M is a sum of hollow submodules and Rad M M 2 Every proper submodule of M is contained in a maximal one and every co nite submoduleof M has a generalized supplement in M 3 M is an irredundant sum of local modules and Rad M M and we prove that a ringR is semilocal if and only if every cyclic right R module is a WGS module 1

محمد رضا ودادي

عاطفه قرباني

منصور معتمدي، احمد حقاني