Commuting anti-homomorphisms

Abstract

ABSTRACTLet R be a semiprime ring with center Z(R) and with extended centroid C. Suppose that τ:R→R is an anti-homomorphism such that the image of τ has small centralizer. It is proved that the following are equivalent: (1) for all x∈R; (2) x+xτ∈Z(R) for all x∈R; (3) xxτ∈Z(R) for all x∈R. In this case, there exists an idempotent e∈C such that (1−e)R is a commutative ring and the semiprime ring eR is equipped with an involution , which is induced canonically by τ. These results generalize both the involution case by Chacron [5] and the anti-automorphism case by the author [7].