On Lie isomorphisms of rings

Abstract

An associative ring $A$ gives rise to the Lie ring $A^{(-)}=(A,[a,b]=ab-ba)$. The subject of isomorphisms of Lie rings $A^{(-)}$ and $[A,A]$ has attracted considerable attention in the literature. We prove that if the identity element of $A$ decomposes into a sum of at least three full orthogonal idempotents, then any isomorphism from the Lie ring $[A,A]$ to the Lie ring $[B,B]$ is standard. For non-unital rings, the description is more intricate. Under a certain assumption on idempotents, we extend a Lie isomorphism from $[A,A]$ to $[B,B]$ to a homomorphism of associative rings $\widehat{A\oplus A^{op}}\to B,$ where $A^{op}=(A,a\cdot b=b\cdot a),$ and $\widehat{A\oplus A^{op}}\to A\oplus A^{op}$ is the universal annihilator extension of the ring $A\oplus A^{op}.$ The results obtained are then applied to the description of automorphisms and derivations of Lie algebras of infinite matrices.