# Jordan Derivations of Special Subrings of Matrix Rings

## TL;DR

This paper characterizes all Jordan derivations of a specific class of matrix rings over a 2-torsion free ring, showing they decompose into derivations and extremal Jordan derivations.

## Contribution

It provides a complete description of Jordan derivations on upper triangular matrix rings with entries in an ideal, extending understanding of their structure.

## Key findings

- Every Jordan derivation decomposes into a derivation plus an extremal Jordan derivation.
- Explicit forms of Jordan derivations are characterized.
- The structure holds for rings with no 2-torsion and identity.

## Abstract

Let $K$ be a 2-torsion free ring with identity and $R_{n}(K,J)$ be the ring of all $n\times n$ matrices over $K$ such that the entries on and above the main diagonal are elements of an ideal $J$ of $K.$ We describe all Jordan derivations of the matrix ring $R_{n}(K,J)$ in this paper. The main result states that every Jordan derivation $\Delta $ of $R_{n}(K,J)$ is of the form $\Delta =D+\Omega $ where $D$ is a derivation of $R_{n}(K,J)$ and $\Omega $ is an extremal Jordan derivation of $R_{n}(K,J).$