RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES


AYDOĞDU P. , Er N., Ertas N. O.

GLASGOW MATHEMATICAL JOURNAL, vol.54, no.3, pp.605-617, 2012 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 54 Issue: 3
  • Publication Date: 2012
  • Doi Number: 10.1017/s0017089512000183
  • Title of Journal : GLASGOW MATHEMATICAL JOURNAL
  • Page Numbers: pp.605-617

Abstract

Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: Acyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (not congruent to R-R) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and boolean AND(n is an element of N)J(n) = J(m) for some m is an element of N if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.