We proved that an exchange ring R the power-substitution property if and only if any one of the following conditions hold: (1) whenever x ∈ R is regular, there exists some positive integer n such that xIn = xWx for some unit-regular element W ∈ Mn(R); (2) whenever x ∈ R is regular, there exist positive integers m, n such that xmIn = xmWxm for some unit-regular element W ∈ Mn(R); (3) whenever x = xyx in R, there exists some positive integer n such that xIn = xyW = Wyx for some unit-regular element W ∈ Mn(R); (4) whenever aR + bR = dR in R, there exist some positive integer n and W, Q ∈ Mn(R), where W is unit-regular, such that aIn + bQ = dW; (5) whenever a1R +···+ akR = dR in R, where k ≥ 1, there exist some positive integer n and unit-regular elements W1, …, Wk ∈ Mn(R) such that a1W1 +···+ akWk = dIn; (6) whenever a1R +···+ akR = dR in R, where k ≥ 1, there exist positive integers m, n and unit-regular elements W1, …, Wk ∈ Mn(R) such that [image omitted]. These results, by replacing the word “unit” with the word “unit-regular, ” generalize the corresponding results of Canfell, Chen, Wu, etc.
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