On rings determined by their idempotents and units

Abstract

This paper describes properties of three certain classes of rings determined by conditions on idempotents and units, namely, the condition that any two generators of each principal right ideal are associated (UG rings), the condition that every principal right ideal is generated by a sum of a unit and an idempotent (P_r ), and the condition xy = 0 implies xsy = 0 for a sum of idempotent and unit s and any elements x, y of a ring (idun-semicommutative rings). It is proved that the class of all UG rings contains every local as well as every von Neumann regular ring, and the condition P_r is satisfied by both semiperfect and regular rings. Both local and abelian regular rings are proved to be necessarily idun-semicommutative. For all three classes are presented some closure properties and illustrating examples.

Keywords:

• (Abelian) regular ring
• clean ring
• semiperfect ring
• semicommutative ring
• symmetric ring

2020 Mathematics Subject Classification:

• 16N20
• 16D60
• 16U60
• 16W10

Acknowledgments

The authors thank the referee for his or her careful reading of the paper and many suggestions leading to a substantial improvement of the paper.