Commutativity preserving maps revisited.
The author refines some results of \textit{K. I. Beidar} and \textit{Y.-F. Lin} [Proc. R. Soc. Edinb., Sect. A, Math. 134, No. 6, 1023-1040 (2004; Zbl 1074.16019)] on the structure of commutativity preserving maps. Given any additive map of rings \(\alpha\colon A\to B\), set \(F(x)=[\alpha(x^2),\alpha(x)]\). The main result considers an additive \(\alpha\colon J\to Q\) where \(J\) is a Jordan ring and \(Q\) is an associative unital algebra with center \(C\), a field with \(\text{char\,}C\neq 2\). The author proves that if \(F(x)=0\) on \(J\), and if \(\alpha(J)\) is 3-free in \(Q\), then either \(\alpha=\mu\beta+\lambda\) for \(\mu\in C\), \(\beta\colon J\to Q\) a Jordan homomorphism, and \(\lambda\colon J\to C\) additive, or else a certain defined ideal of \(J\) is sent into \(C\) by \(\alpha\). By analyzing this ideal of \(J\) when \(J\) is associative, some corollaries of the main result are obtained. One such is that for \(A\) and \(B\) prime rings, neither of \(\text{char\,}2\) and neither satisfying the standard identity \(S_4\), then any additive and surjective \(\alpha\colon A\to B\) with \(F(A)=0\) must be \(\alpha=\mu\beta+\lambda\), as above, or \(\alpha(I)=0\) for a nonzero ideal \(I\) of \(A\). When \(A\) and \(B\) are centrally closed over \(C\) with \(\text{char\,}C\neq 2\), \(A\) does not satisfy \(S_4\), and \(\alpha\) is bijective and linear with \(F(A)=0\), then again \(\alpha=\mu\beta+\lambda\). The first of these corollaries has a version for \(A\) and \(B\) prime with involution and \(\alpha\) surjective with \(F(S(A))=0\) for \(S(A)\) the set of symmetric elements in \(A\).
scientific article; zbMATH DE number 5279146
scientific article