Two questions of L. Va${\rm \check{\textbf{s}}}$ on *-clean rings

TL;DR

This paper investigates the properties of *-clean rings, answering two open questions by providing examples and characterizations related to the structure of *-clean, clean, and regular rings.

Contribution

It answers two open questions about *-clean rings with examples and offers new characterizations of unit regular and *-regular rings.

Findings

Existence of clean but not *-clean rings demonstrated.

Unit regular and *-regular rings are characterized.

Several examples illustrating the properties of *-clean rings.

Abstract

A ring $R$ with an involution * is called (strongly) *-clean if every element of $R$ is the sum of a unit and a projection (that commute). All *-clean rings are clean. Va${\rm \check{s}}$ [L. Va${\rm \check{s}}$, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra 324 (12) (2010) 3388-3400] asked whether there exists a *-ring that is clean but not *-clean and whether a unit regular and *-regular ring is strongly *-clean. In this paper, we answer both questions by several examples. Moreover, some characterizations of unit regular and *-regular rings are provided.