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چكيده انگليسي :

Uniform almost relative injective modules Mahboube Bakhshande m bakhshande@math iut ac ir 2019 Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mahmood Behboodi mbehboodi@cc iut ac irAdvisor Dr Mohammad Reza Vedadi mrvedadi@cc iut ac irMSC 16D50 16E50 Keywords Almost relative injectives Almost self injectives CSmodules VonNeumannregular rings Abstract Uniform almost relative injective modules This M Sc thesis is based on the following papers S SJournal of Algebra 478 2017 353 366The concept of a module M being almost N injective where N is a module was introduced by Baba 1989 Mis said to be almost N injective if for any homomorphism f A M A N either f extends to a homo morphism g N M or there exist a decomposition N N1 N2 with N1 0 and a homomorphismh M N1 such that hf x x for any x A where N N1 is a projection with kernel N2 This concept plays a significant role is studying extending modules Amodule M that is almost M injective is calledan almost self injective module For a given module M the class of modules N for which M is almost N injective is not closed under direct sums Baba gave a necessary and sufficient condition under which a uniform finite lengthmodule U is almost V injective where V is a finite direct sum of uniform finite length modules in terms of extendingproperties of simple submodules of V Let U be a uniform module and V be a finite direct sum of indecomposablemodules Recently Singh 2016 has determined some conditions under which U is almost V injective whichgeneralize Baba s result In the present thesis some more results in this direction are proved Let M be an almostself injective uniform modules Let T be the set of those maximal homomorphisms f N M N Mwhich cannot be extended to endomorphisms of M In Section3 of Chapter 3 an algebraic structure on T is given As defined in Chapter 4 A module M is said to be completely almost self injective if for any two subfactors A Bof M A is almost B injective A necessary and sufficient condition for a module M to be completely almost self injective is given Using this it is proved that a Von Neumann ring R is completely almost right self injective if and Ronly if soc R is semi simple and every minimal right ideal of R is injective We have the following main result RTheorem 1 Let MR be a uniform module and NR be any module Then M is almost N injective if and only if forany maximal homomorphism f L M L N which cannot be extended from N to M N N1 ker fsuch that f N1 L M Theorem 2 Let MR be a uniform module and NR N1 N2 where Ni are indecom posable Then M is almostN injective if and only if one of the followingholds i M is N1 injective and N2 injective ii M is almost N1 injective and almost N2 injective If M is neither N1 injective nor N2 injective then N is aCS module Theorem 3 Let MR be a completely almost self injective module Let S be a simple module which is a subfactor ofM Then either S is M injective or there exists a decom position M M1 M2 such that M2 is a complementof S and M1 is uniserial and has length2 If J M 0 then S is M injective Theorem 4 Let M be a completely almost self injective module Then i Soc M e M

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