Derivations and 2-local derivations on matrix algebras over commutative algebras

Abstract

We characterize derivations and 2-local derivations from $M_n(\mathcal{A})$ into $M_n(\mathcal{M})$, $n\ge 2$, where $\mathcal{A}$ is a unital algebra over $\mathbb{C}$ and $\mathcal{M}$ is a unital $\mathcal{A}$-bimodule. We show that every derivation $D:M_n(\mathcal{A})\to M_n(\mathcal{M})$, $n\ge 2,$ is the sum of an inner derivation and a derivation induced by a derivation from $\mathcal{A}$ to $\mathcal{M}$. We say that $\mathcal{A}$ commutes with $\mathcal{M}$ if $am=ma$ for every $a\in \mathcal{A}$ and $m\in \mathcal{M}$. If $\mathcal{A}$ commutes with $\mathcal{M}$ we prove that every inner 2-local derivation $D:M_n(\mathcal{A})\to M_n(\mathcal{M})$, $n\ge 2$, is an inner derivation. In addition, if $\mathcal{A}$ is commutative and commutes with $\mathcal{M}$, then every 2-local derivation $D:M_n(\mathcal{A})\to M_n(\mathcal{M})$, $n\ge 2$, is a derivation.