Uniquely strongly clean triangular matrices
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A ring R is uniquely (strongly) clean provided that for any a is an element of R there exists a unique idempotent e is an element of R (e is an element of comm(a)) such that a e is an element of U(R). We prove, in this note, that a ring R is uniquely clean and uniquely bleached if and only if R is abelian, T-n(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 T-n(R) is uniquely strongly clean for some n >= 1. In the commutative case, more explicit results are obtained.