## Turkish Journal of Mathematics

#### DOI

10.3906/mat-0906-60

#### 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

#### First Page

399

#### Last Page

404

#### Recommended Citation

ALI, FAISAL and CHAUDHRY, MUHAMMAD ANWAR
(2011)
"On generalized (\alpha,\beta)-derivations of semiprime rings,"
*Turkish Journal of Mathematics*: Vol. 35:
No.
3, Article 5.
https://doi.org/10.3906/mat-0906-60

Available at:
https://journals.tubitak.gov.tr/math/vol35/iss3/5