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 T_n(R) of all n×n upper triangular matrices over R is a left derivation, and any generalized Jordan left derivation on T_n(R) is a generalized left derivation. Moreover, we prove that any generalized left derivation on T_n(R) is decomposed into the sum of a right centralizer and a Jordan left derivation. Some related results are also obtained.