Uniquely strongly clean triangular matrices

Abstract

A ring R is uniquely (strongly) clean provided that for any a ∈ R there exists a unique idempotent e ∈ R ( e ∈ comm(a) ) such that a − e ∈ U(R). We prove, in this note, that a ring R is uniquely clean and uniquely bleached if and only if R is abelian, Tn(R) is uniquely strongly clean for all n ≥ 1, i.e. every n x n triangular matrix over R is uniquely strongly clean, if and only if R is abelian, and Tn(R) is uniquely strongly clean for some n ≥ 1. In the commutative case, more explicit results are obtained.A ring R is uniquely (strongly) clean provided that for any a ∈ R there exists a unique idempotent e ∈ R ( e ∈ comm(a) ) such that a − e ∈ U(R). We prove, in this note, that a ring R is uniquely clean and uniquely bleached if and only if R is abelian, Tn(R) is uniquely strongly clean for all n ≥ 1, i.e. every n x n triangular matrix over R is uniquely strongly clean, if and only if R is abelian, and Tn(R) is uniquely strongly clean for some n ≥ 1. In the commutative case, more explicit results are obtained.