**Author(s): ** Shuliang Huang**Journal: ** Boletim da Sociedade Paranaense de Matemática ISSN 0037-8712

**Volume: ** 29;

**Issue: ** 1;

**Start page: ** 9;

**Date: ** 2011;

Original page**Keywords: ** sigma-prime ring |

derivation |

generalized derivation |

(alpha |

beta )-derivation |

commutativity.**ABSTRACT**

Let R be a 2-torsion free sigma-prime ring with an involution sigma, I a nonzero sigma-ideal of R. In this paper we explore the commutativity of R satisfying any one of the properties: (i)d(x)◦F(y) = 0 for all x, y ∈ I. (ii) [d(x),F(y)] = 0 for all x,y ∈ I. (iii) d(x)◦F(y) = x◦y for all x, y ∈ I. (iv) d(x)F(y) − xy ∈ Z(R) for all x, y ∈ I. We also discuss (alpha,beta)-derivations of sigma-prime rings and prove that if G is an (alpha,beta)-derivation which acts as a homomorphism or as an anti-homomorphism on I, then G = 0 or G = on I.

## You may be interested in: