A class of uniquely (strongly) clean rings

In this paper we call a ring R δ_r -clean if every element is the sum of an idempotent and an element in δ(R_R) where δ(R_R) 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) δ_r -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 δ_r -clean rings lies between the class of uniquely clean rings and the class of exchange rings, and the class of uniquely strongly δ_r -clean rings is a subclass of the class of uniquely strongly clean rings. We prove that R is δ_r -clean if and only if R/δ_r(R_R) is Boolean and R/Soc(R_R) is clean where Soc(R_R) is the right socle of R.