Journals / Turkish Journal of Mathematics / 2011 / Cilt: 35 - Sayı: 3
On generalized (a,b)-derivations of semiprime rings
- Pages
- 399–404
- DOI
- —
Abstract
We investigate some properties of generalized (a,b)-derivations on semiprime rings. Among some other results, we show that if g is a generalized (a,b)-derivation, with associated (a,b)-derivation d, on a semiprime ring R such that [g(x),a(x)]=0 for all x\in R, then d(x)[y,z]=0 for all x,y,z\in R and d is central. We also show that if a,n,t are endomorphisms and b,m are automorphisms of a semiprime ring R and if R has a generalized (a,b)-derivation g, with associated (a,b)-derivation d, such that g([m(x),w(y)])=[n(x),w(y)]a,t, where w:R\rightarrow R is commutativity preserving, then [y,z]d(w(p))=0 for all y,z,p\in R.
Özet
We investigate some properties of generalized (a,b)-derivations on semiprime rings. Among some other results, we show that if g is a generalized (a,b)-derivation, with associated (a,b)-derivation d, on a semiprime ring R such that [g(x),a(x)]=0 for all x\in R, then d(x)[y,z]=0 for all x,y,z\in R and d is central. We also show that if a,n,t are endomorphisms and b,m are automorphisms of a semiprime ring R and if R has a generalized (a,b)-derivation g, with associated (a,b)-derivation d, such that g([m(x),w(y)])=[n(x),w(y)]a,t, where w:R\rightarrow R is commutativity preserving, then [y,z]d(w(p))=0 for all y,z,p\in R.