On strongly~*-clean rings

摘要

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. Va?, ~*-Clean rings; some clean and almost clean Baer ~*-rings and von Neumann algebras, J. Algebra 324 (2010) 33883400]. 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. Va?, ~*-Clean rings; some clean and almost clean Baer ~*-rings and von Neumann algebras, J. Algebra 324 (2010) 33883400] are answered.