Commutativity criteria for prime rings with involution via pairs of endomorphisms

The aim of this article is to investigate central-valued identities involving pairs of endomorphisms on prime rings equipped with an involution of the second kind. Extending the recent contributions of Mir et al. (2020) and Boua et al. (2024), we establish several new commutativity criteria for such rings in the presence of two distinct nontrivial endomorphisms. Our approach provides a unified technique that covers multiple classes of $\ast$-identities and yields generalizations of earlier single-endomorphism results. Moreover, explicit counterexamples are constructed to demonstrate the necessity of the hypotheses on primeness and on the nature of the involution.

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