On rings with associated elements

Kosan M. T., Truong Cong Quynh T. C. Q., Sahinkaya S.

COMMUNICATIONS IN ALGEBRA, vol.45, no.7, pp.2747-2756, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 45 Issue: 7
  • Publication Date: 2017
  • Doi Number: 10.1080/00927872.2016.1175595
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2747-2756
  • Gazi University Affiliated: No


A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for all a, b in a ring R, aR = bR implies a = bu for some unit u of R). In the present paper, we study "uniquely generated modules" as a module theoretic version of "uniquely generated ideals," and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results: End(M) is unit-regular if and only if End(M) is regular and all M-cyclic submodules of a right R-module M are uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ring R[x; sigma]/(x(n+1)), where R is a strongly regular ring with an endomorphism sigma be an endomorphism of R.