2025/08/10 by Brešar, Matej, Zelmanov, Efim · 2 citations
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2508.07191
Let A and B be associative algebras over a field F with \rm char(F)≠ 2. Our first main result states that if A is unital and equal to its commutator ideal, then every Jordan epimorphism φ:A→ 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∈ A | a^*=a\. We show that there exists a \rm T-ideal G having the following two properties: (1) the Jordan homomorphism φ:H(G(A),*)→ B can be extended to an (associative) homomorphism, subject to the condition that the subalgebra generated by φ(H(A,*)) has trivial annihilator, and (2) every element of the \rm T-ideal of identities of the algebra of 2× 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.