A NOTE ON STRONGLY CLEAN MATRICES

Abstract

A ring R is strongly clean provided that for any a ∈ R, there exist an idempotent e ∈ R and a unit u ∈ R such that a = e + u and eu = ue. Let T3(R) be a special subring of 3 by 3 matrix ring over R. We prove, in this article, that T3(R) is strongly clean if and only if for any a ∈ J(R), b ∈ R, c ∈ 1 + J(R), either lb − ra or lb − rc is surjective. Similar characterization is obtained for T3(R) over a weak h-ring R.