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Commutativity criteria for prime rings with involution via pairs of endomorphisms

Gurninder Singh Sandhu, Geetika Gudwani, Mohammadi El Hamdoui

TL;DR

The paper addresses when a prime ring $R$ with an involution $\ast$ of the second kind becomes commutative under central-valued identities involving two endomorphisms $g_1,g_2$. It develops a unified method to handle a family of $\ast$-identities, extending previous results for a single endomorphism to the two-endomorphism setting and generalizing Mir et al. and Boua et al. The authors prove that, for a $2$-torsion-free prime ring with a second-kind involution, any of the identities $A_1$–$A_4$ (and related variants) forces commutativity unless the endomorphisms preserve the center, and they confirm a two-endomorphism conjecture while providing counterexamples to show the necessity of primeness and a second-kind involution. The results yield a cohesive framework for commutativity criteria in noncommutative ring theory and broaden the applicability of endomorphism-driven identities in rings with involution, with explicit constructions illustrating the sharpness of hypotheses.

Abstract

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.

Commutativity criteria for prime rings with involution via pairs of endomorphisms

TL;DR

The paper addresses when a prime ring with an involution of the second kind becomes commutative under central-valued identities involving two endomorphisms . It develops a unified method to handle a family of -identities, extending previous results for a single endomorphism to the two-endomorphism setting and generalizing Mir et al. and Boua et al. The authors prove that, for a -torsion-free prime ring with a second-kind involution, any of the identities – (and related variants) forces commutativity unless the endomorphisms preserve the center, and they confirm a two-endomorphism conjecture while providing counterexamples to show the necessity of primeness and a second-kind involution. The results yield a cohesive framework for commutativity criteria in noncommutative ring theory and broaden the applicability of endomorphism-driven identities in rings with involution, with explicit constructions illustrating the sharpness of hypotheses.

Abstract

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 -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.
Paper Structure (2 sections, 11 theorems, 68 equations)

This paper contains 2 sections, 11 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Lemma 2.5

Let $R$ be a 2-torsion free prime ring with involution $\ast$ of the second kind. If $R$ admits nontrivial endomorphisms $g_{1},g_{2}$ satisfying any of $(A_{1})$-$(A_{4})$, then either $R$ is commutative or $g_{1}(Z(R))\subseteq Z(R)$ and $g_{2}(Z(R))\subseteq Z(R).$

Theorems & Definitions (25)

  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • proof
  • Lemma 2.8
  • proof
  • Theorem 2.9
  • proof
  • ...and 15 more