Neşet Aydin

Copyright © 1997 Neşet Aydin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let R be a prime ring of characteristic not 2, U a nonzero ideal of R and 0≠da(α,β)-derivation of R where α and β are automorphisms of R. i) [d(U),a]=0 then a∈Z ii) For a,b∈R, the following conditions are equivalent (I) α(a)d(x)=d(x)β(b), for all x∈U (II) Either α(a)=β(b)∈CR(d(U)) or CR(a)=CR(b)=R′ and a[a,x]=[a,x]b (or a[b,x]=[b,x]b) for all x∈U. Let R be a 2-torsion free semiprime ring and U be a nonzero ideal of R iii) Let d be a (α,β)-derivation of R and g be a (γ,δ)-derivation of R. Suppose that dg is a (αγ,βδ)-derivation and g commutes both γ and δ then g(x)Uα−1d(y)=0, for all x,y∈U iv) Let Ann(U)=0 and d be an (α,β)-derivation of Rand g be a (λ,δ)-derivation of R such that g commutes both γ, and δ. If for all x,y∈U, β−1(d(x))Ug(y)=0=g(x)Uα−1(d(y)) then dg is a (αγ,βδ)-derivation on R.