On generalized semiderivations in 3-prime near-rings

Abstract

There is a large body of evidence showing that the existence of a suitably-constrained derivation on a 3-prime near-ring forces the near-ring to be a commutative ring. The purpose of this paper is to study generalized semiderivations which satisfy certain identities on 3-prime near-ring and generalize some results due to [H. E. Bell and G. Mason, On derivations in near-rings, North-Holland Math. Stud. 137 (1987) 31–35; H. E. Bell, On prime near-rings with generalized derivation, Int. J. Math. Math. Sci. 2008 (2008), Article ID: 490316, 5[Formula: see text]pp; A. Boua and L. Oukhtite, Some conditions under which near-rings are rings, Southeast Asian Bull. Math. 37 (2013) 325–331]. Moreover, an example is given to prove that the necessity of the 3-primeness hypothesis imposed on the various theorems cannot be marginalized.