International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 4, Pages 813-816
Research note

Notes on (α,β)-derivations

Neşet Aydin

Adnan Menderes University, Faculty of Arts and Sciences, Department of Mathematics, Aydin 0910, Turkey

Received 18 December 1995; Revised 2 April 1996

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.


Let R be a prime ring of characteristic not 2, U a nonzero ideal of R and 0da(α,β)-derivation of R where α and β are automorphisms of R. i) [d(U),a]=0 then aZ ii) For a,bR, the following conditions are equivalent (I) α(a)d(x)=d(x)β(b), for all xU (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 xU. 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)Uα1d(y)=0, for all x,yU 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,yU, β1(d(x))Ug(y)=0=g(x)Uα1(d(y)) then dg is a (αγ,βδ)-derivation on R.