作为同态与反同态的广义(θ,θ)-导子的研究

The Research of Generalized(θ,θ)-derivation as Homomorphism and Anti-homomorphism

R是2-扭自由素环,I是R上的非零理想,θ是R上的自同构,F是R上的与(θ,θ)-导子d有关的非零广义(θ,θ)-导子,有F(xy)=F(x)F(y)或F(xy)=F(y)F(x),对所有的x,y属于I且d≠0,则R是可交换的.

The concept of(θ,θ)-derivations as well as generalized(θ,θ)-derivations have been generalized as an additive function F:R→R satisfying F(xy)=F(x)θ(y)+θ(x)d(y)for all x,y∈R,where d is a nonzero(θ,θ)-derivation on R.Such a function F is said to be a generalized(θ,θ)-derivation.In the present paper it is shown that:If R is 2-torsion free prime ring,I≠0 an ideal of R and F is a generalized(θ,θ)-derivation of R such that either F(xy)=F(x)F(y)or F(xy)=F(y)F(x)for all x,y∈I,then R is commutative.