On internally cancellable rings

ALTUN M., Ozcan C.

JOURNAL OF ALGEBRA AND ITS APPLICATIONS, vol.16, no.6, 2017 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 16 Issue: 6
  • Publication Date: 2017
  • Doi Number: 10.1142/s0219498817501171
  • Journal Name: JOURNAL OF ALGEBRA AND ITS APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Keywords: Unit-regular element, (idempotent) stable range one, internal cancellation, perspectivity, special clean element, STABLE RANGE, SUBSTITUTION
  • Hacettepe University Affiliated: Yes

Abstract

The concept of internally cancellable rings has been extensively studied in the literature. This paper seeks to continue the study of these rings and find some new characterizations. It is proved that R is "IC", if and only if for each regular element a is an element of R, and idempotent element b is an element of R with Ra + Rb = R, there exists x is an element of R such that a + xb is a unit (alternatively, unit-regular element) in R and aR boolean AND xR = 0. In case the ring R has the summand sum property, we indicate that R is IC, if and only if for each regular element a is an element of R, and element b is an element of R with Ra + Rb = R, there exists an idempotent a is an element of R, such that a vertical bar b is a unit in R and aR boolean AND eR = 0.