Lie (Jordan) centralizers on generalized matrix algebras

Abstract

Let be a commutative ring with unit element, be -algebras, be an -bimodule, and be a -bimodule. The -algebra is a generalized matrix algebra defined by the Morita context In this article, we study Lie (Jordan) centralizer on generalized matrix algebras and obtain the necessary and sufficient conditions for a Lie centralizer map to be proper. Further, we prove that every Jordan centralizer is a centralizer on generalized matrix algebras under certain assumptions.