A NOTE ON STRONGLY CLEAN MATRICES

Authors : Huanyin Chen

Pages : 192-204

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Publication Date : 2011-12-01

Article Type : Research Paper

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 T3insert ignore into journalissuearticles values(R); be a special subring of 3 by 3 matrix ring over R. We prove, in this article, that T3insert ignore into journalissuearticles values(R); is strongly clean if and only if for any a ∈ Jinsert ignore into journalissuearticles values(R);, b ∈ R, c ∈ 1 + Jinsert ignore into journalissuearticles values(R);, either lb − ra or lb − rc is surjective. Similar characterization is obtained for T3insert ignore into journalissuearticles values(R); over a weak h-ring R.
Keywords : strongly clean ring, matrix ring, local ring