σ-Commuting and σ-Centralizing Anti-homomorphisms

Let R be a semiprime ring with center Z(R) and with extended centroid C and let σ : R → Rbe an automorphism. Assume that τ : R → Ris an anti-homomorphism, such that the image of τ has small centralizer. It is proved that the following are equivalent: (1) xσ xτ = xτ xσ for all x ∈ R; (2) xσ + xτ ∈ Z(R) for all x ∈ R; (3) xσ xτ ∈ 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 τ , which is induced canonically by τ . Note that one can easily obtained the main result in Lee (Commun Algebra 46(3):1060–1065, 2018) when σ = idR.