Document Type : Research Paper
Academic member of University of Kurdistan
Abstract. 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.