6626

6177

كارشناسي ارشد

رياضي محض ﴿جبر﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1390

[هشت]، 86ص

ص.ع. به فارسي و انگليسي

1396/10/12

كتابنامه

علوم رياضي

رياضي

ID6177

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

Injectivity Relative to Closed Submodules Seyed Abollhassan Sepehr Hosseini sa sepehrhosseini@math iut ac ir January 28 2012 Master of Science Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Atefeh Ghorbani a ghorbani@cc iut ac irAdvisor Dr Mahmood Behboodi m behboodi@cc iut ac ir2000 MSC 16D50Key words P pure injective modules c injective modules almost principal ideals Dedekind do mains Abstract In this thesis all rings are associative with identity and all modules are unitary left modules LetR be any ring A submodule K of an R module M is called closed in M provided K has no properessential extension in M Clearly every direct summand of M is closed in M Moreover if L is anysubmodule of M then there exists by Zorn s Lemma a submodule K of M maximal with respectto the property that L is essential submodule of K and this case K is a closed submodule of M A module M is called an extending module if every closed submodule is a direct summand An R module X is called M c injective provided for every closed submodule K of M every homomorphism K X can be lifted to a homomorphism M X Morever X is called c injective providedX is M c injective for every R module M Note that if M is an extending module then every R module is M c injective If R is a Dedekind domain and an R module M is a direct product of simpleR modules then M is M c injective We prove that if R is a Dedekind domain then an R moduleX is c injective if and only if there exists an R module Y such that Y is a direct product of simpleR modules and injective R modules with the property that X is isomorphic to a direct summand ofY We show that such a direct summand is isomorphic to a direct product of homogeneous semisimpleR modules and injective R modules Let P is collection of left primitive ideals of R A submodule Lof an R module M is called P pure in M provided L IM IL for every ideal I in P An R moduleX is called P pure injective provided for every R module M and every P pure submodule L of M every homomorphism L X can be lifted to a homomorphism M X We rst characterizeP pure injective modules over rings R such that R P is an Artinian ring for every left primitive idealP of R Commutative rings clearly have this property More generally rings satisfying a polynomialidentity FBN rings and semiperfect rings satisfy this property Then show that for a Dedekind domainthe class of c injective modules is precisely the class of P pure injective modules We show that thecharacterization does not extend to commutative Noetherian domains which are not Dedekind Infact we prove that if R is a commutative Noetherian domain and P a maximal ideal of R then thesimple R module R P is c injective if and only if the ideal P is invertible Finally we show that acommutative Noetherian domain R is Dedekind if and only if every simple R module is c injective

عاطفه قرباني

محمود بهبودي

منصور معتمدي، احمد حقاني