11229

872 دكتري

دكتري

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1394

نه، 69ص.

به فارسي و انگليسي

1395/02/07

رياضي

ID872 دكتري

Rings Whose Certain Modules are Direct Sums of Cyclic Modules Mahdieh Naji Esfahani m najyesfahani@math iut ac ir 2016 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Atefeh Ghorbani a ghorbani@cc iut ac ir Advisor Dr Mahmood Behboodi mbehbood@cc iut ac ir 2010 MSC 16D70 16U80 16P50 16P40 13F10 03E10 Keywords FGC ring Cyclic module Principal ideal ring Duo ring Noetherian module Coatomicmodule Local dimension Local module AbstractMany studies have been conducted to characterize commutative rings whose nitely generated modulesare direct sums of cyclic modules called FGC rings however the characterization of noncommutativeFGC rings is still an open problem even for duo rings We study FGC rings in some special cases itis shown that a local Noetherian ring R is FGC if and only if R is a principal ideal ring if and onlyif R is a uniserial ring and if these assertions hold R is a duo ring We characterize Noetherian duoFGC rings In fact it is shown that a duo ring R is a Noetherian left FGC ring if and only if R is aNoetherian right FGC ring if and only if R is a principal ideal ring Another family of rings whose certain modules are direct sums of cyclic modules is the family ofrings which ideals are direct sums of cyclic modules These rings were previously studied only incommutative local case To study intended rings we de ne and study local dimension In general wede ne local dimension for coatomic modules Next we study commutative rings with countable localdimension which ideals are direct sums of cyclic modules It is shown that for a commutative ring R l dim R and every ideal of R is a direct sum of cyclic modules if and only if R is a non localprincipal ideal domain which is not local or R F V such that F is a eld and V is a discretevaluation domain

عاطفه قرباني

محمود بهبودي

احمد حقاني، عليرضا نصراصفهاني، محمدرضا ودادي