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 \(\widehat{A\oplus A^{\textrm{op}}}\rightarrow B,\) where \(A^{\textrm{op}}=(A, a\cdot b= b\cdot a),\) and \(\widehat{A\oplus A^{\textrm{op}}}\rightarrow A\oplus A^{\textrm{op}}\) is the universal annihilator extension of the ring \(A\oplus A^{\textrm{op}}.\) The results obtained are then applied to the description of automorphisms and derivations of Lie algebras of infinite matrices.
