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

TL;DR

This paper characterizes derivations and 2-local derivations on matrix algebras over commutative algebras, showing conditions under which they are inner or actual derivations, extending understanding of algebraic structures.

Contribution

It provides a comprehensive characterization of derivations and 2-local derivations on matrix algebras over commutative algebras, including conditions for their innerness.

Findings

Every derivation is a sum of an inner derivation and one induced by a derivation on the base algebra.

Inner 2-local derivations are shown to be inner derivations under certain commutation conditions.

All 2-local derivations are derivations when the base algebra is commutative and commutes with the bimodule.

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.