Jordan Derivations of Special Subrings of Matrix Rings

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 $R_{n}(K,J).$