A class of uniquely (strongly) clean rings

In this paper we call a ring R δr -clean if every element is the sum of an idempotent and an element in δ(RR) where δ(RR) is the intersection of all essential maximal right ideals of R. If this representation is unique (and the elements commute) for every element we call the ring uniquely (strongly) δr -clean. Various basic characterizations and properties of these rings are proved, and many extensions are investigated and many examples are given. In particular, we see that the class of δr -clean rings lies between the class of uniquely clean rings and the class of exchange rings, and the class of uniquely strongly δr -clean rings is a subclass of the class of uniquely strongly clean rings. We prove that R is δr -clean if and only if R/δr (RR) is Boolean and R/Soc(RR) is clean where Soc(RR) is the right socle of R.