On generalized (\alpha,\beta)-derivations of semiprime rings

Authors: FAISAL ALI, MUHAMMAD ANWAR CHAUDHRY

Abstract: We investigate some properties of generalized (\alpha,\beta)-derivations on semiprime rings. Among some other results, we show that if g is a generalized (\alpha,\beta)-derivation, with associated (\alpha,\beta)-derivation \delta, on a semiprime ring R such that [g(x),\alpha(x)]=0 for all x\in R, then \delta(x)[y,z]=0 for all x,y,z\in R and \delta is central. We also show that if \alpha,\nu,\tau are endomorphisms and \beta,\mu are automorphisms of a semiprime ring R and if R has a generalized (\alpha,\beta)-derivation g, with associated (\alpha,\beta)-derivation \delta, such that g([\mu(x),w(y)])=[\nu(x),w(y)]_{\alpha,\tau}, where w:R\rightarrow R is commutativity preserving, then [y,z]\delta(w(p))=0 for all y,z,p\in R.

Keywords: Semiprime ring, derivation, generalized derivation, generalized (\alpha,\beta)-derivation

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