Power-Substitution and Exchange Rings

Abstract

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 xI n = xWx for some unit-regular element W ∈ M n (R); (2) whenever x ∈ R is regular, there exist positive integers m, n such that x m I n = x m Wx m for some unit-regular element W ∈ M n (R); (3) whenever x = xyx in R, there exists some positive integer n such that xI n = xyW = Wyx for some unit-regular element W ∈ M n (R); (4) whenever aR + bR = dR in R, there exist some positive integer n and W, Q ∈ M n (R), where W is unit-regular, such that aI n + bQ = dW; (5) whenever a 1 R +···+ a k R = dR in R, where k ≥ 1, there exist some positive integer n and unit-regular elements W 1, …, W k ∈ M n (R) such that a 1 W 1 +···+ a k W k = dI n ; (6) whenever a 1 R +···+ a k R = dR in R, where k ≥ 1, there exist positive integers m, n and unit-regular elements W 1, …, W k ∈ M n (R) such that . These results, by replacing the word “unit” with the word “unit-regular, ” generalize the corresponding results of Canfell, Chen, Wu, etc.