A ring R is uniquely (strongly) clean provided that for any a ∈ R there exists a unique idempotent e ∈ R ∈ comm(a) such that a − e ∈ U (R). Let R be a uniquely bleached ring. We prove, in this note, that R is uniquely clean if and only if R is abelian, and Tn(R) is uniquely strongly clean for all n ≥ 1, if and only if R is abelian, Tn(R) is uniquely strongly clean for some n ≥ 1. In the commutative case, the more explicit results are obtained. These also generalize the main theorems in [6] and [7], and provide many new class of such rings.

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Let R be an associative ring with identity. An element a ∈ R is called clean if a = e + u with e an idempotent and u a unit of R and a is called strongly clean if, in addition, eu = ue. A ring R is called clean if every element of R is clean and R is strongly clean if every element of R is strongly clean. In the paper [Nicholson and Zhou, Clean rings: a survey, Advances in Ring Theory, 181-198, World Sci. Pub., Hackensack, NJ, 2005], the authors brought out an up to date account of the results in the study of clean rings. Here, we give an account of the results on strongly clean rings.

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