Lie isomorphisms of prime rings satisfying \(St_4\) (Q1849294)

Let \(R\) and \(R'\) be prime rings with respective extended centroids \(C\) and \(C'\), and central closures \(R_C\) and \(R_C'\). An additive \(T\colon R\to R'\) is a Lie map if \(T([x,y])=[T(x) T(y)]\) for all \(x,y\in R\), where \([x,y]=xy-yx\). The main result proves that when \(\text{char }R\neq 2\) then any Lie isomorphism from \(R\) to \(R'\) is the sum of a monomorphism or negative of an anti-monomorphism \(f\colon R\to R_C'\) and an additive \(g\colon R\to C'\) satisfying \(g([R,R])=0\). This result extends a similar one of \textit{M. Brešar} [Trans. Am. Math. Soc. 335, No. 2, 525-546 (1993; Zbl 0791.16028)] that required the restriction than neither \(R\) nor \(R'\) embeds in some \(M_2(F)\) for a field \(F\).