Jordan derivations of some classes of matrix rings

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 ail upper triangular matrices over R; that is, T-n(R) subset of S subset of 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. Jondrup. 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).