Very cleanness of generalized matrices

Abstract

An element a in a ring R is very clean in case there exists an idempotent e 2 R such that ae = ea and either a 􀀀 e or a + e is invertible. An element a in a ring R is very J-clean provided that there exists an idempotent e 2 R such that ae = ea and either a􀀀e 2 J(R) or a + e 2 J(R). Let R be a local ring, and let s 2 C(R). We prove that A 2 Ks(R) is very clean if and only if A 2 U(Ks(R)); I A 2 U(Ks(R)) or A 2 Ks(R) is very J-clean.