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

Abstract

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