ON STRONGLY *-CLEAN RINGS

A *-ring R is called a *-clean ring if every element of R is the sum of a unit and a projection, and R is called a strongly *-clean ring if every element of R is the sum of a unit and a projection that commute with each other. These concepts were introduced and discussed recently by [L. Vas, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra 324 (2010) 3388-3400]. Here it is proved that a *-ring R is strongly *-clean if and only if R is an abelian, *-clean ring if and only if R is a clean ring such that every idempotent is a projection. As consequences, various examples of strongly *-clean rings are constructed and, in particular, two questions raised in [L. Vas, *-Clean rings; some clean and almost clean Baer *-rings and von Neumann algebras, J. Algebra 324 (2010) 3388-3400] are answered.