# Two-Local derivations on associative and Jordan matrix rings over commutative rings

## TL;DR

This paper proves that 2-local inner derivations on matrix rings over commutative rings are actually inner derivations, and extends these results to Jordan matrix rings, establishing their derivation properties.

## Contribution

It establishes that 2-local inner derivations on matrix rings over commutative rings are inner, and develops a Jordan analog for such derivations.

## Key findings

- Every 2-local inner derivation on matrix rings over commutative rings is an inner derivation.
- Every derivation on an associative ring extends to its matrix ring.
- Every 2-local inner derivation on Jordan matrix rings over commutative rings is a derivation.

## Abstract

In the present paper we prove that every 2-local inner derivation on the matrix ring over a commutative ring is an inner derivation and every derivation on an associative ring has an extension to a derivation on the matrix ring over this associative ring. We also develop a Jordan analog of the above method and prove that every 2-local inner derivation on the Jordan matrix ring over a commutative ring is a derivation.