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چكيده انگليسي :

Primary decomposition for left Noetherian ring Zohreh Karimi zohreh karimi@math iut ac ir January 17 2015 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mahmood Behboodi mbehbood@cc iut ac irAdvisor Dr Hossein Khabazian khabaz@cc iut ac ir2000 MSC 16P40 16P70Keywords primary decomposition uniform decomposition the left spectrum left primes uni form module prime M idealAbstract This thesis is an extension and a generalization of the work done byV V Bavula Primary Decompositions for Left Noetherian Rings Algebr Represent Theor 13 2010 103 118 In mathematics the Lasker Noether theorem states that every Noetherian ring is a Lasker ring which means that every ideal can be written as an intersection of finitely many primary ideals Ithas a straightforward extension to modules stating that every submodule of a finitely generatedmodule over a Noetherian ring is a finite intersection of primary submodules This contains thecase for rings as a special case considering the ring as a module over itself so that ideals aresubmodules Let R be a commutative Noetherian ring M an R module and N a submodule of M Aprimary decomposition of N is a representation of the form N Y1 Yn where each Yi is aprimary submodule of M If R is a commutative Noetherian ring and M is a finitely generatedR module Then any proper submodule N of M has a primary decomposition In this paper primary decomposition of a submodule of a finitely generated module over acommutative Noetherian ring was generalized for modules over a not necessarily commutative left Noetherian ring First we introduce the left prime spectrum of a ring that is a natural generalization of the spectrumin the commutative situation and an analogue of associated primes so called associated leftprimes for modules over noncommutative rings The set of all associated left primes of M isdenoted by AsR M Next we mention one other way of generalizing primary decomposition to left Noetherian rings A submodule N of a module M is primary if M N has finite uniform dimension and the set AsR M N consists of a single element we say also that the intersection N s Ni is ai 1primary decomposition of N if each Ni is a primary submodule of M Next the notions of primarysubmodule and primary decomposition of a Noetherian module will be generalized to a largerfamily of modules We call this family the family of uniformly finite modules In particular weshow that for an arbitrary ring R the primary decomposition theory holds for any R module Mwhich has the property that each factor module of M is finite dimensional in the sense of Goldie and we show that the main properties of primary decompositions for commutative Noetherianrings still hold in this situation Also we introduce the shortest primary decomposition and the maximal shortest primary de composition of a submodule of a uniformly finite module and we show that for a submoduleof a uniformly finite module a shortest primary decomposition always exists and each shortestprimary decomposition is contained in a maximal shortest primary decomposition

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تجزيه اوليه براي حلقه هاي نوتري چپ

تجزيه يكنواخت , طيف چپ , اول هاي چپ , مدول يكنواخت , M- ايده آل اول