Jordan homomorphisms and T-ideals

Brešar, Matej (author), Zelmanov, Efim (author)

Let $A$ and $B$ be associative algebras over a field $F$ with ${\rm char}(F)\ne 2$. Our first main result states that if $A$ is unital and equal to its commutator ideal, then every Jordan epimorphism $\varphi:A\to B$ is the sum of a homomorphism and an antihomomorphism. Our second main result concerns (not necessarily surjective) Jordan homomorphisms from $H(A,*)$ to $B$, where $*$ is an involution on $A$ and $H(A,*)=\{a\in A\,|\, a^*=a\}$. We show that there exists a ${\rm T}$-ideal $G$ having the following two properties: (1) the Jordan homomorphism $\varphi:H(G(A),*)\to B$ can be extended to an (associative) homomorphism, subject to the condition that the subalgebra generated by $\varphi(H(A,*))$ has trivial annihilator, and (2) every element of the ${\rm T}$-ideal of identities of the algebra of $2\times 2$ matrices is nilpotent modulo $G$. A similar statement is true for Jordan homomorphisms from $A$ to $B$. A counter-example shows that the assumption on trivial annihilator cannot be removed.

• Language: English
• Keywords: Jordan homomorphisms, trivial annihilator, T-ideals
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.02.2026
• Year: 2026
• Number of pages: 24 str.
• Numbering: Vol. 113, iss. 2, article no. e70448
• PID: 20.500.12556/RUL-181514
• UDC: 512
• ISSN on article: 0024-6107
• DOI: 10.1112/jlms.70448
• COBISS.SI-ID: 274702595
• Publication date in RUL: 09.04.2026

Record is a part of a journal

• Title: Journal of the London Mathematical Society
• Shortened title: J. Lond. Math. Soc.
• Publisher: Wiley, London Mathematical Society
• ISSN: 0024-6107
• COBISS.SI-ID: 5188362

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• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.