Let U be an algebra over a unital commutative ring R, and let σ : U → U be an algebraic homomorphism. In this paper, we consider a linear map L : U → U that satisfies one of the following conditions: a, b ∈ U, ab = 0 ⇒ L([a, b]) = [L(a), σ(b)]; a, b ∈ U, ab = 0 ⇒ L([a, b]) = [σ(a), L(b)]; or a, b ∈ U, ab = 0 ⇒ L([a, b]) = [L(a), σ(b)] = [σ(a), L(b)], where [a, b] = ab − ba is the Lie product in U. We characterize linear maps L under automorphisms σ by reducing to the case σ = idU (the identity map on U). Using these equivalences, we further characterize the linear maps L on generalized matrix algebras; triangular algebras; von Neumann algebras; standard operator algebras and nest algebras for every automorphism σ on these algebras. We also obtain similar results for Lie σ-centralizers. Some of our results generalize some of the previous results.
