On idempotents in relation with regularity

We make a study of two generalizations of regular rings, concentrating our attention on the structure of idempotents. A ring R is said to be right attaching-idempotent if for there exists such that ab is an idempotent. Next R is said to be generalized regular if for there exist nonzero such that ab is a nonzero idempotent. It is first checked that generalized regular is left-right symmetric but right attaching-idempotent is not. The generalized regularity is shown to be a Morita invariant property. More structural properties of these two concepts are also investigated.