2013/09/22 by Ghahramani, Hoger
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1309.5570
Let \Mn be the ring of all n × n matrices over a unital ring R, let M be a 2-torsion free unital \Mn-bimodule and let D:\Mn→ M be an additive map. We prove that if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)=0 whenever \A,\B∈ \Mn are such that \A\B=\B\A=0, then D(\A)=δ(\A)+\A D(1), where δ:\Mn→ M is a derivation and D(1) lies in the centre of M. It is also shown that D is a generalized derivation if and only if D(\A)\B+ \A D(\B)+D(\B)\A+ \B D(\A)-\A D(1)\B-\B D(1)\A=0 whenever \A\B=\B\A=0. We apply this results to provide that any (generalized) Jordan derivation from \Mn into a 2-torsion free \Mn-bimodule (not necessarily unital) is a (generalized) derivation. Also, we show that if φ:\Mn→ \Mn is an additive map satisfying φ(\A \B+\B \A)=\Aφ(\B)+φ(\B)\A (\A,\B ∈ \Mn), then φ(\A)=\Aφ(1) for all \A∈ \Mn, where φ(1) lies in the centre of \Mn. By applying this result we obtain that every Jordan derivation of the trivial extension of \Mn by \Mn is a derivation.