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چكيده انگليسي :

T Semisimple Modules And T Semisimple Rings MOHAMMAD HOSSEIN LAVAEY mh lavaey@math iut ac ir September 18 2014 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mohammad Reza Vedadi mr vedadi@cc iut ac irAdvisor Mr Hossein Khabazian khabaz@cc iut ac ir2010 MSC 16D10 16D70 16D90 16P70Keywords Nonsingular and Z2 torsion modules T essential submodules T semisimple submodulesAbstract This M Sc thesis is based on the following paper Shadi Asgari Ahmad Haghany Yaser Tolooei T SemisimpleModules And T Semisimple Rings Communications in Algebra 41 5 2013 1882 1902 We define and investigate t semisimple modules as a generalization of semisimple modules A module M is calledt semisimple if every submodule N M contains a direct summand K of M such that K is t essential in N We will show in Theorem 2 3 3 Corollaries 2 3 7 2 3 11 and Proposition 2 3 13 that t semisimple modules admitmany other characterizations Submodules and homomorphic images of t semisimple modules inherit the property and every direct sum of t semisimple modules is again a t semisimple module We will show that there exists alargest t semisimple submodule in any module M Z2 M S M where Z2 M is the Goldie torsion submoduleand S M is the sum of nonsingular simple submodule of M It will be shown that a semilocal module M is t semisimple if and only if Rad M is Z2 torsion The t semisimple property is invariant under Morita equivalences T semisimple modules form a strict subclass of t extending modules In Section 3 1 we defined and study right t semisimple rings A ring R is called right t semisimple if RR is t semisimple Every right Artinian local ring is right t semisimple Various characterizations of right t semisimplerings are given Accordingly a t semisimple ring is exactly a direct product of two rings one is semisimple and theother is right Z2 torsion For rings being t semisimple is not a symmetric property and we give an example of a right t semisimple ring which is not left t semisimple We determine rings R for which the classes of semisimple R modules t semisimple R modules and t extending R modules coincide For some types of rings conditions equivalent to beingt semisimple are found Section 3 2 concerns with other characterizations of right t semisimple rings relative to chain conditions It is shownthat a ring R is right t semisimple if and only if every nonsingular R module has ACC ascending chain condition respectively DCC descending chain condition on essential submodules The ring R is right t semisimple if andonly if Rad R is Z2 torsion and R Z2 RR satisfies DCC on principal right ideals As a consequence of this weobtain an equivalent condition to being right t semisimple for a ring which is either right continuous or left quasi dualor left small dual Moreover we see that every quasi frobenius ring is right t semisimple For rings we have quasi frobenius right t semisimple right t extendingbut none of these implications is reversible Another result of interest which tightly connects the FGF conjecturewith properties of a single Z2 torsion module A ring R is right t semisimple if and only if every nonsingularcyclic resp nonsingular finitely generated right ideal R is a direct summand of RR and R satisfies ACC on rightideals which contain Z2 RR We conclude that a right self injective ring R is right t semisimple if and only ifR Z2 RR is a right Noetherian ring Finally we give an example of a ring R for which every nonsingular cyclicR module is injective but it is not right t semisimple that is not every nonsingular R module is injective

لوايي، محمدحسين

حلقه ها و مدول هاي t- نيمساده

مدول هاي نامنفرد و Z2- تابدار , زير مدول هاي t- اساسي