14205

1295 دكتري

دكتري

رياضي

اصفهان : دانشگاه صنعتي اصفهان

1397

نه، [۱۰۵]ص.: مصور

محمود بهبودي

امير هاشمي

فارسي به انگليسي; انگليسي به فارسي

احمد موسوي، جواد اسداللهي

1397/10/10

كتابنامه

رياضي

رياضي

ID1295 دكتري

Almost uniserial and almost serial rings and modules Samira Roointan s roointan@math iut ac ir August 19 2015 Doctor of Philosophy Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Mahmood Behboodi mbehbood@cc iut ac ir Advisor Dr Amir Hashemi amir hashemi@cc iut ac ir Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran AbstractWe study the class of almost uniserial rings as a straightforward common generalization ofleft uniserial rings and left principal ideal domains A ring R is called almost left uniserial ifany two non isomorphic left ideals of R are linearly ordered by inclusion i e for every pairI J of left ideals of R either I J or J I or I J Also an R module M is called almost uniserial if any two non isomorphic submodules are linearly ordered by inclusion Wegive some properties of almost uniserial rings and modules It is shown that a left almostuniserial ring is either a local ring or its maximal left ideals are cyclic A Noetherian leftalmost uniserial ring is a local ring or a principal left ideal ring Also a left Artinian princi pal left ideal ring R is almost left uniserial if and only if R is left uniserial or R M2 D where D is a division ring We also consider Artinian commutative rings which are almostuniserial and obtain a structure theorem for these rings In the sequel we obtain some resultswhich are inspired by a characterization of Artinian serial rings as rings having all left orright modules serial We prove that if R is a local ring and all left R modules are almostserial then R is an Artinian ring which is uniserial either on the left or on the right Also aconnection between local rings having all left and right modules almost serial local balancedrings studied by Dlab and Ringel and local K the rings have been produced We also prove oMorita invariance of the almost serial property and list some consequences In continuingthe study of decomposition aspects one natural question is that if we weaken the assump tion of almost seriality of all modules and just assume that each ideal is almost serial whatwill be the structure of the ring This question is completely answered for commutative rings Key Words Uniserial ring uniserial module almost uniserial ring almost uniserial module almost serial ring almost serial module left principal ideal ring K the ring balanced ring oMSC 2010 13A18 13F30 16D20 1

محمود بهبودي

امير هاشمي

احمد موسوي، جواد اسداللهي