A Note on 2x2 Uniquely Clean Idempotent Matrices and Strongly Clean Matrices

Abstract

An element of a ring is said to be clean if it is the sum of a unit and an idempotent. It is uniquely clean if this representation is unique. It is called strongly clean if it is the sum of an idempotent and a unit that commute. It is well known that central idempotents in any ring are uniquely clean [1]. In this note it has been shown that the converse is also true in M₂ (R) , R an Integral Domain. When R is a projective free ring, a characterization of strongly clean elements in M_n (R) has been given [2]. When R is a principal ideal domain (P.I.D.), towards such a characterization we take an approach which uses well known structure of idempotent matrices in M_n (R) . We use this to characterize non triangular strongly clean elements in M₂ (Z) in terms of their entries.