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Lie isomorphisms of prime rings satisfying $St_4$. (English) Zbl 1009.16033
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 $\text{char }R\ne 2$ 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 {\it M. BreŇ°ar} [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 $F$.

16W10Associative rings with involution, etc.
16N60Prime and semiprime associative rings
16W20Automorphisms and endomorphisms of associative rings
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