$$\sigma $$-Commuting and $$\sigma $$-Centralizing Anti-homomorphisms

Let $${\mathcal {R}}$$ be a semiprime ring with center $$Z({\mathcal {R}})$$ and with extended centroid C and let $$\sigma : {\mathcal {R}} \rightarrow {\mathcal {R}}$$ be an automorphism. Assume that $$\tau : {\mathcal {R}} \rightarrow {\mathcal {R}} $$ is an anti-homomorphism, such that the image of $$\tau $$ has small centralizer. It is proved that the following are equivalent: (1) $$x^{\sigma }x^{\tau } = x^{\tau }x^{\sigma }$$ for all $$x\in {\mathcal {R}};$$ (2) $$x^{\sigma } + x^{\tau }\in Z({\mathcal {R}})$$ for all $$x\in {\mathcal {R}};$$ (3) $$x^{\sigma }x^{\tau }\in Z({\mathcal {R}})$$ for all $$x\in {\mathcal {R}}.$$ In this case, there exists an idempotent $$e \in C$$ , such that $$(1-e){\mathcal {R}}$$ is a commutative ring and the semiprime ring $$e{\mathcal {R}}$$ is equipped with an involution $$\widetilde{\tau }$$ , which is induced canonically by $$\tau $$ . Note that one can easily obtained the main result in Lee (Commun Algebra 46(3):1060–1065, 2018) when $$\sigma =id_{{\mathcal {R}}}.$$