5010

4714

كارشناسي ارشد

رياضي محض﴿ جبر﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1388

[شش] ، 65ص.

ص.ع. به: فارسي و انگليسي

محمد رضا ودادي

حسين خبازيان

19/11/88

محمود بهبودي، عاطفه قرباني

رياضي

ID4714

به فارسي و انگليسي: قابل رويت در نسخه ديجيتال

Ojective Modules Azadeh Dogmechin a dogmechin@math iut ac ir Nowember 02 2009 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Mohmmad Reza Vedadi mrvedadi@cc iut ac ir2000 MSC 16D50 16D70 16D80 Key words Ojective Semi injective Exchangable Abstract In this thesis we present an expanded account of Ojective Modules based on an articleby Saad H Mohamed and Bruo J Muller 2002 A module is extending or a CS module or a module with C1 if every submodule is essential ina direct summand equivalently if every closed submodule is a direct summand Extendingmodules generalize injective and quasi continuous ones they have been studied extensivelybut many questions remain unresolved The most annoying open problem is to nd sensible necessary and su cient conditions forthe direct sum of extending modules to be extending Mutual injectivity is su cient but notnecessary claim that a weakened version of mutual injectivity is equivalent to a strengthenedversion of C1 as follows Any closed submodule C of M M1 Mn satis es M C M1 Mn with Mi Mi if and only if the Mi are extending and Mj ojective for allj i A proof of this claim is apparently not available The present paper gives such a proof at least for n 2 or for uniform Mi The generalcase remains open but a slightly stronger version of mutual ojectivity will do We alsoo er several related results for instance that ojectivity passes to direct summands Whetherojectivity passes to direct sums remains open and is related to the di culties for n 2alluded to above We use the notation A e M and B M to indicate that A is anessential submodule and B is a direct summand of M Summand will mean direct summand A closed submodule of M is one which has no essential extensions in M The graph of ahomomorphism X Y is the submodule x x x X of X Y ForM i I Mi and K I M K is de nd as M K i K Mi We show that if M M1 Mn where the Mi are uniform Then M has C1 and thedecomposition is exchangeable if and only if Mi is Mj ojective for all i j We show that if M M1 M2 Then M is semi continuous if and only if Mi is semi continuous and Mj ojective for j i We show that M M1 Mn where the Mi areuniform It is prove that a module M is semi continuous if and only if M has C1 and forevery decomposition M A B A and B are mutually ojective 1

محمد رضا ودادي

حسين خبازيان

محمود بهبودي، عاطفه قرباني