5154

4828

كارشناسي ارشد

رياضي

اصفهان: دانشگاه صنعتي اصفهان،دانشكده علوم رياضي

1388

[هفت]،83ص.

ص.ع.به فارسي و انگليسي

حسين خبازيان

محمود بهبودي

21/2/89

جواد اسداللهي،محمدرضا ودادي

رياضي

ID4828

به فارسي و انگليسي:قابل رويت در نسخه ديجيتالي

FI extending Modules and Rings Hamid Reza Mahmoodi hr mahmoodi@math iut ac ir January 12 2010 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Hossein khbazian Khabaz@cc iut ac ir 2000 MSC Primary 13C10 16D10 Secondary 16D50 Key words direct summand essential extending FI extending fully invariant submodule ideal module ring submodule Abstract In recent years the theory of extending modules and rings has come to play an important role in theory of rings and modules A module M is called extending if every submodule of M is essential in a direct summand In this thesis we call a module FI extending if every fully invariant submodule is essential in a direct summand One advantage of this generalization of the extending property over various other generalizations is that the underpinnings i e the fully invariant submodules form a complete modular sublattice of the lattic of submodules and are well behaved with respect to endomorphisms The class of fully invariant submodules includes many of the most signi cant submodules of a module e g the Jacobson radical the socle the singular submodule etc Initially we develop basic properties in the module setting For example in contrast to extending modules a direct sum of FI extending mod ules is FI extending but direct product of FI extending modules need not be FI extending Later we largely focus on the speci c case when a ring is FI extending considered as a module over itself In section 2 we show that every right left FI extending ring has a maximal nonsingular FI extending direct summand which is also a right left nonsingular right left FI extending FI extending ring We prove that if R is FI extending then M n R and Tn R are so for any n 1 We obtain a generalized triangular matrix representation for rings which are FI extending on both sides In fact if R is left and right FI extending then R1 R12 R13 R T 0 R2 R23 0 0 R3 where each Rij is a Ri Rj bimodule Proposition 4 19 Also we prove that if R is left and right FI extending then R R 1 R2 where R1 is a reduced ring and R2 is a ring in which every nonzero ideal contains a nonzero nilpotent element of R We investigate the interconnections e g extending quasi extending Baer quasi Baer continuous quasi continuous etc Finally we obtain a ring decomposition for a right left nonsingular right left FI extending ring using the concept of an orthogonal pair of modules classes 1

حسين خبازيان

محمود بهبودي

جواد اسداللهي،محمدرضا ودادي