Strong commutativity preserving maps in prime rings with involution. (Q1044580)

This paper is one of a number that studies functions preserving commutativity on special subsets of prime rings. Let \(R\) be a prime ring with involution, \(\text{char\,}R\neq 2\), center \(Z(R)\), and set of symmetric elements \(S\). The main result in this paper considers an additive \(g\colon S\to R\) satisfying \([g(s),g(t)]=[s,t]\) for all \(s,t\in S\) and proves that \(R\) embeds in \(M_2(F)\) for a field \(F\), or there is an additive \(\mu\colon S\to Z(R)\) so that \(S=\{s\in S\mid g(s)=s+\mu(s)\}\) or \(S=\{s\in S\mid g(s)=-s+\mu(s)\}\).