7284

6790

كارشناسي ارشد

رياض محض﴿جبر﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1391

ده، 91ص.

ص.ع. به فارسي و انگليسي

1396/09/21

كتابنامه

علوم رياضي

رياضي

ID6790

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

Chain Conditions on Non summands Zahra Jafari z jafari@math iut ac ir 17 09 2012 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Mahmood Behboodi mbehboodi@cc iut ac ir Supervisor Dr Hossein Khabazian hkhabazian@cc iut ac ir 2010 MSC 16A53 16A64 Keywords Ascending chain condition Descending chain condition Summand Semisimplemodules Noetherian rings Artinian rings Essential submodules Dedekind domain AbstractLet R be any ring This thesis is based on the works done on 5 and 6 it is shown that modulessatisfying ascending or descending chain conditions resp acc and dcc on non summand submodulesbelongs to some particular classes such as the class of all R modules nitely generated nitedimensional and cyclic modules are considered It is proved that a module M satis es ascending resp descending chain condition on non summands if and only if M is semisimple or Noetherian resp Artinian Over a right Noetherian ring R a right R module M satis es ascending chaincondition on nitely generated non summands if and only if M satis es ascending chain conditionon non summands a right R module M satis es descending chain condition on nitely generatednon summands if and only if M is locally Artinian Moreover if a ring R satis es descending chaincondition on cyclic non summand right ideals then R is a semiregular ring such that the Jacobsonradical J is left t nilpotent We shall give an example of a commutative von Neumann regular ringR such that R satis es ascending chain condition and descending chain condition on essential idealsand on super uous ideals but R satis es neither ascending nor descending chain conditions on non summands For an in nite cardinal number and M a unital right module over a ring R we show that every wellordered ascending resp descending chain of essential submodules of M has cardinality less than ifand only if every well ordered ascending respectively descending chain of submodules of M soc M has cardinality less than We use this to show that a CS module with an chain condition onessential submodules is a direct sum of a module with that same chain condition on all submodulesand a semisimple module Thus a CS module with fewer than generators has an chain condition

محمود بهبودي، حسين خبازيان

منصور معتمدي، جواد اسدالهي