5007

4711

كارشناسي ارشد

رياضي محض﴿جبر﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1388

[هشت] ، 70ص: مصور، جدول، نمودار

ص.ع. به: فارسي و انگليسي

محمد رضا ودادي

محمود بهبودي

19/11/88

منصور معتمدي، حسين خبازيان

رياضي

ID4711

به فارسي و انگليسي: قابل رويت در نسخه ديجيتال

Radical Submodules and Uniform Dimension of Modules Fatemeh Rashedi f rashedi@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 16N60 16P60 16E10 16D80 Key words Irreducible prime submodule Radical submodule Prime decomposition Uniform dimension Fully bounded goldie Abstract In this thesis we present an expanded account of Radical Submodules and UniformDimension of Modules based on an article by P F Smith 2004 Throughout this note all rings are associative with identity and all modules are unital leftmodules Let R be a ring and let M be an R module A submodule K of M is called primeif K M and whenever r R and L is a submodule of M such that rL K then rM Kor L K In this case the ideal P r R rM K is a prime ideal of R and we call Ka P prime submodule of M A submodule N of a module M is called a radical submoduleif N is an intersection of prime submodules of M Note that radical submodules are propersubmodules of M Given a submodule N of a module M a decomposition N K1 Kn interms of submodules Ki 1 i n of M where n is a positive integer is called irredundantif N K1 Ki 1 Ki 1 Kn for all 1 i n A submodule N of a module Mis said to have a prime decomposition if N is the intersection of a nite collection of primesubmodules of M Let N be a submodule of an R module M such that N has a primedecomposition Then N will be said to have a normal prime decomposition if there existsa positive integer n distinct prime ideals Pi 1 i n of R and Pi prime submodules Ki 1 i n of M such that N K1 Kn is an irredundant decomposition We investigate the relations between a radical submodule N of a module M being a niteintersection of prime submodules of M and the factor module M N having nite uniformdimension It is proved that if N is a radical submodule of a module M over a ring R suchthat M N has nite uniform dimension then N is a nite intersection of prime submodules The converse is false in general but is true if the ring R is fully left bounded left Goldie andthe module M is nitely generated It is further proved that in general if a submodule Nof a module M is a nite intersection of prime submodules then the module M N can havean in nite number of minimal prime submodules It is prove that if zero is a radical submodule of a module M over a ring R Then the followingstatements are equivalent i The zero submodule of M is a nite intersection of irreducibleprime submodules of M ii M has nite uniform dimension 1

محمد رضا ودادي

محمود بهبودي

منصور معتمدي، حسين خبازيان