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.
Categories: math.RA math.OA