On Jordan left derivations and generalized Jordan left derivations of matrix rings

Let R be a 2-torsion free ring with identity. In this paper, first we prove that any Jordan left derivation (hence, any left derivation) on the full matrix ringMn(R) (n  2) is identically zero, and any generalized left derivation on this ring is a right centralizer. Next, we show that if R is also a prime ring and n  1, then any Jordan left derivation on the ring Tn(R) of all n×n upper triangular matrices over R is a left derivation, and any generalized Jordan left derivation on Tn(R) is a generalized left derivation. Moreover, we prove that any generalized left derivation on Tn(R) is decomposed into the sum of a right centralizer and a Jordan left derivation. Some related results are also obtained.