On Regular Power-Substitution

The necessary and sufficient conditions under which a ring satisfies regular power-substitution are investigated. It is shown that a ring R satisfies regular power-substitution if and only if a≂b in R implies that there exist n ∈ ℕ and a U ∈ GL _n (R) such that aU = Ub if and only if for any regular x ∈ R there exist m, n ∈ ℕ and U ∈ GL _n (R) such that x ^m I _n = x ^m Ux ^m , where a≂b means that there exists x, y, z ∈ R such that a = ybx, b = xaz and x = xyx = xzx. It is proved that every directly finite simple ring satisfies regular power-substitution. Some applications for stably free R-modules are also obtained. Project supported by the grant of Hangzhou Normal University (No. 200901).