On Clean Ideals

We introduce the notion of clean ideal, which is a natural generalization of clean rings. It is shown that every matrix ideal over a clean ideal of a ring is clean. Also we prove that every ideal having stable range one of a regular ring is clean. These generalize the corresponding results for clean rings. 1. Introduction. Let R be a unital ring. We say that R is a clean ring in case every element of R is a sum of an idempotent and a unit. It is well known that every endomorphism ring of a countably generated vector space over a division ring is a clean ring (cf. [7, Theorem]). A ring R is said to be unit regular in case for every x ∈ R, there exists a unit u ∈ R such that x = xux. Answering a question of Nicholson, Camillo and Yu [3, Theorem 5] claimed that every unit regular ring is clean. But there was a gap in their proof. In [2, Theorem 1], Camillo and Khurana proved this result by a new route and gave a characterization of unit regular rings. In this paper, a natural problem asked whether there is a nonclean ring R while some element of R is a sum of an idempotent and a unit. So as to deal with such rings, we will introduce a notion of clean ideals. We show that clean ideals have similar properties to clean rings. Throughout, all rings are associative with identity and all modules are right modules. We always use U(R) to denote the set of units of R.