On Jordan $ \sigma $-centralizers and related linear maps in algebras

Nura Alotaibi; Nura Alotaibi

doi:10.3934/math.2026184

AIMS Mathematics

2026, Volume 11, Issue 2: 4571-4585. doi: 10.3934/math.2026184

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Research article

On Jordan $ \sigma $-centralizers and related linear maps in algebras

Nura Alotaibi ^,

Mathematics Department, College of Science, Jouf University, Sakaka P. O. Box 2014, Saudi Arabia

Received: 18 December 2025 Revised: 02 February 2026 Accepted: 04 February 2026 Published: 24 February 2026
MSC : 7B40, 15A78, 46L10, 47L10, 47L35

Let $ \mathcal{B} $ be an algebra over a commutative ring with identity $ S $, and let $ \sigma $: $ \mathcal{B}\to\mathcal{B} $ be an algebra homomorphism. In this paper, we study linear operators $ \Delta $: $ \mathcal{B}\to\mathcal{B} $ that are constrained by zero-product conditions involving the Jordan product $ u\circ v = uv + vu $. In particular, we consider mappings that satisfy
$ uv = 0 \Rightarrow \Delta(u\circ v) = \Delta(u)\circ \sigma(v), \; \; \; uv = 0 \Rightarrow \Delta(u\circ v) = \sigma(u)\circ \Delta(v), $
and
$ uv = 0 \Rightarrow \Delta(u\circ v) = \Delta(u)\circ\sigma(v) = \sigma(u)\circ\Delta(v). $
Assuming the endomorphism $ \sigma $ is bijective, we prove that the scenario essentially simplifies to the identity case $ \sigma = \mathrm{id}_{\mathcal{B}} $. Such a simplification enables a comprehensive the forms of these linear operators. As a result, we obtain precise expressions for these operators across diverse algebraic structures, including generalized matrix algebras, upper-triangular algebras, von Neumann algebras, standard operator algebras, and nest algebras. Moreover, this approach produces analogous results for Jordan $ \sigma $-centralizers, thus extending and integrating various prior findings in the field.
- Jordan $ \sigma $-centralizers,
- zero-product,
- algebras isomorphism,
- generalized matrix algebras,
- upper-triangular algebras,
- von Neumann algebras,
- standard operator algebras,
- nest algebras
Citation: Nura Alotaibi. On Jordan $ \sigma $-centralizers and related linear maps in algebras[J]. AIMS Mathematics, 2026, 11(2): 4571-4585. doi: 10.3934/math.2026184

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Abstract

Let $ \mathcal{B} $ be an algebra over a commutative ring with identity $ S $, and let $ \sigma $: $ \mathcal{B}\to\mathcal{B} $ be an algebra homomorphism. In this paper, we study linear operators $ \Delta $: $ \mathcal{B}\to\mathcal{B} $ that are constrained by zero-product conditions involving the Jordan product $ u\circ v = uv + vu $. In particular, we consider mappings that satisfy

$ uv = 0 \Rightarrow \Delta(u\circ v) = \Delta(u)\circ \sigma(v), \; \; \; uv = 0 \Rightarrow \Delta(u\circ v) = \sigma(u)\circ \Delta(v), $

and

$ uv = 0 \Rightarrow \Delta(u\circ v) = \Delta(u)\circ\sigma(v) = \sigma(u)\circ\Delta(v). $

Assuming the endomorphism $ \sigma $ is bijective, we prove that the scenario essentially simplifies to the identity case $ \sigma = \mathrm{id}_{\mathcal{B}} $. Such a simplification enables a comprehensive the forms of these linear operators. As a result, we obtain precise expressions for these operators across diverse algebraic structures, including generalized matrix algebras, upper-triangular algebras, von Neumann algebras, standard operator algebras, and nest algebras. Moreover, this approach produces analogous results for Jordan $ \sigma $-centralizers, thus extending and integrating various prior findings in the field.

References

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