Jordan elementary maps on rings.
There are many results in the literature showing that maps on rings that satisfy certain multiplicative conditions are necessarily additive. The paper continues this line of investigation. Specifically, the main result states that if \(R\) is a prime ring such that \(\text{char}(R)\neq 2\) and \(R\) contains a nontrivial idempotent, then surjective maps \(M\colon R\to R'\) and \(M^*\colon R'\to R\), where \(R'\) is an arbitrary ring, must be additive in case they satisfy the conditions \(M(xM^*(y)x)=M(x)yM(x)\) and \(M^*(yM(x)y)=M^*(y)xM^*(y)\) for all \(x\in R\), \(y\in R'\). These conditions appear in the definition of a Jordan elementary operator in the paper [Acta Sci. Math. 66, No. 3-4, 769-791 (2000; Zbl 0973.47028)] by \textit{M. Brešar}, \textit{L. Molnár} and \textit{P. Šemrl}.
scientific article; zbMATH DE number 2081913