STRONGLY CLEAN MATRICES OVER COMMUTATIVE LOCAL RINGS

Abstract

An element of a ring is called strongly clean provided that it can be written as the sum of an idempotent and a unit that commute. We characterize, in this paper, the strongly cleanness of matrices over commutative local rings. This partially extend many known results such as Theorem 12 in Borooah, Diesl and Dorsey [Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (2008) 281-296], Theorem 3.2.7 and Proposition 3.3.6 in Dorsey [Cleanness and strong cleanness of rings of matrices, Ph.D. thesis, University of California, Berkeley (2006)], Theorem 2.3.14 in Fan [Algebraic analysis of some strongly clean and their generalization, Ph. D. thesis, Memorial University of Newfoundland, Newfoundland (2009)], Theorem 3.1.9 and Theorem 3.1.26 in Yang [Strongly clean rings and g(x)-clean rings, Ph.D. thesis, Memorial University of Newfoundland, Newfoundland (2007)].