JORDAN DERIVATIONS OF SOME CLASSES OF MATRIX RINGS

Abstract

Let $R$ be a 2-torsionfree ring with identity and $S$ be a subring of the ring $M_n(R)$ that contains the ring $T_n(R)$ of all upper triangular matrices over $R$; that is, $T_n(R) \subseteq S \subseteq M_n(R)$. The goal of this paper is to describe a Jordan derivation $\Delta$ on $S$. The main result states that $\Delta$ can be uniquely represented as the sum of a derivation and a very special Jordan derivation. This result describes also the structure of every derivation on the ring $S$ which is an extension of a result of S.P. Coelho and C.P. Milies and a result of $S$. Jøndrup. Moreover, one of the corollaries of the main theorem covers the classical result by Jacobson and Rickart stating that there are no proper Jordan derivations on $M_n(R)$, and a more recent result by D. Benkovic that there are no proper Jordan derivations on the algebra $T_n(R)$.