$(\sigma, \tau)$-derivations on prime near rings

Mohammad Ashraf, Asma Ali and Shakir Ali

 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

E-mail. mashraf80@hotmail.com

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example [1], [2], [3], [4], [5] and [8]) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like behaviour. In this paper we generalize some of the results due to Bell and Mason [4] on near-rings admitting a special type of derivation namely $(\sigma,\tau)$- derivation where $\sigma,\tau$ are automorphisms of the near-ring. Finally, it is shown that under appropriate additional hypothesis a near-ring must be a commutative ring.

AMSclassification. 16W25, 16Y30.

Keywords.  prime near-ring, derivation, $\sigma$-derivation, $(\sigma, \tau)$-derivation, $(\sigma, \tau)$-commuting derivation.