Rings in which every element is the sum of a left zero-divisor and an idempotent


Ghashghaei E., KOŞAN M. T.

PUBLICATIONES MATHEMATICAE-DEBRECEN, vol.95, pp.321-334, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 95
  • Publication Date: 2019
  • Doi Number: 10.5486/pmd.2019.8410
  • Journal Name: PUBLICATIONES MATHEMATICAE-DEBRECEN
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.321-334
  • Keywords: idempotent, nil-clean, zero-clean, zero-divisor
  • Gazi University Affiliated: Yes

Abstract

A ring R is called left zero clean if every element is the sum of a left zero-divisor and an idempotent. This class of rings is a natural generalization of O-rings and nil-clean rings. We determine when a skew polynomial ring is a left zero-clean ring. It is proved that a ring R is left zero-clean if and only if the upper triangular matrix ring T-n(R) is left zero-clean. It is shown that a commutative ring R is zero-clean if and only if the matrix ring M-n(R) is zero-clean for every positive integer n >= 1. We characterize the zero-clean matrix rings over fields. We also determine when a 2 x 2 matrix A over a field is left zero-clean. A ring is called uniquely left zero-clean if every element is uniquely the sum of a left zero-divisor and an idempotent. We completely determine when a ring is uniquely left zero-clean.