# Local derivations on associative and Jordan matrix algebras

## TL;DR

This paper proves that local derivations on matrix and Jordan matrix algebras over various fields and rings are actually derivations, extending known results to broader algebraic structures.

## Contribution

It establishes that all additive local inner derivations on matrix and Jordan matrix algebras are genuine derivations, even over arbitrary fields and certain finite rings.

## Key findings

- Local derivations are derivations in matrix algebras over arbitrary fields.
- Local derivations are derivations in Jordan algebras of symmetric matrices over arbitrary fields.
- The results extend to matrix rings over finite rings generated by the identity or integers.

## Abstract

In the present paper we prove that every additive (not necessarily homogenous) local inner derivation on the algebra of matrices over an arbitrary field is an inner derivation, and every local inner derivation on the ring of matrices over a finite ring generated by the identity element or the ring of integers is an inner derivation. We also prove that every additive local inner derivation on the Jordan algebra of symmetric matrices over an arbitrary field is a derivation, and every local inner derivation on the Jordan ring of symmetric matrices over a finite ring generated by the identity element or the ring of integers is a derivation.