حلقه هايي كه در آن هر ايدآل تصويري

چكيده انگليسي :

Rings is which every ideal is pure projective or FP projective Azin zalekian a zalekian@mah iut ac ir July 16 2018 Master of Science Thesis Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Mahmood Behboodi m behboodi@cc iut ac irAdvisor Dr Bijan Taeri b taeri@cc iut ac ir2000 MSC 16E60 16D40 13C10Keywords Pure projevtive F P projevtive Pure hereditary F P hereditary F P injective RD hereditary Abstract This M Sc thesis is based on the following paper Rings in which every ideal is pure projective or F P projevtive A M D S H S Journal of Algebra 478 419 436 2017 According to Warfiel s criterion pure hereditary modules can be defined as direct summands of direct sums offinitely presented modules A left R module M is called F P injective or absolutely pure if it is pure in every leftR module that contains it A left R module P is called F P projective if ExtR 1 P M 0 A ring R is said tobe left hereditary if every ideal of R is projective A ring R is said to be left pure hereditary resp RD hereditary if every left ideal of R is pure projective resp RD projective In this thesis some properties and examples of these rings which are nontrivial generalization of hereditary rings are given First it is shown that the notion of pure hereditay rings and RD hereditary rings are nontrivial general ization of hereditary rings pure semisimple rings Notherian rings and principal ideal ring A ring R is said to be leftF P hereditary if every left ideal of R is F P projective F L Sandomierski proved that if R is a left hereditary ring then R is left Noetherian if and only if u dim R R In one of the theorems it is generalized Sandomirski stheorem In fact it is shown that if R is a left RD hereditary nonsingular ring then R is left Noerherian if and onlyif u dim R R A ring R is said to be a left p p ring resp left F P ring if every principal left ideal of R is projective resp flat A semiprime ideal in a ring R is any ideal of R which is an intersection of prime ideals A semiprime ring isany ring in which 0 is a semiprime ideal A ring R is said to be Goldie if R has finite uniform dimension and theascending chain condition on its annihilators In this thesis we have the following important theorems and propositions Proposition 1 The following conditions are equivalent for a ring R 1 R is a left hereditary ring 2 R is a left RD hereditary and left semi hereditary ring 3 R is a left pure hereditary and left semi hereditary ring 4 R is a left RD hereditary ring and every left ideal of R is flat 5 R is a left pure hereditary ring and every left ideal of R is flat Recall that a ring R is called left perfect if every left R module has a projective cover A ring R is local in caseR has an unique maximal ideal A ring R is semilocal if R J R is a semisimple Artinian ring A left duo ring isa ring in which every left ideal is two sided Also a ring R is said to be a strongly regular ring if R satisfies thed c c on chains of the form rR r2 R r3 R that is for every r R there exist t 1 and s R such that r t r t 1 s This is a right left symmetric condition

ذالكيان، آذين

حلقه هايي كه در آن هر ايدآل تصويري - محض يا FP- تصويري است

حلقه هاي موروثي , تصويري محض , FP- تصويري , FP- تزريقي , FP- موروثي