Two questions of L. Va$\check{\text{s}}$ on *-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$\check{\mathrm{s}}$ [L. Va$\check{\mathrm{s}}$, *-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.