A class of uniquely (strongly) clean rings


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

Keywords: Clean ring, strongly clean ring, uniquely clean ring, strongly J-clean ring

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