Prime near-ring modules and their links with the generalised group near-ring

摘要

In view of the facts that the definition of a ring led to the definition of a near- ring, the definition of a ring module led to the definition of a near-ring module, prime rings resulted in investigations with respect to primeness in near-rings, one is naturally inclined to attempt to define the notion of a group near-ring seeing that the group ring had already been defined and investigated into by, interalia, Groenewald in [7] . However, in trying to define the group near-ring along the same lines as the group ring was defined, it was found that the resulting multiplication was, in general, not associative in the near-ring case due to the lack of one distributive property. In 1976, Meldrum [19] achieved success in defining the group near-ring. How- ever, in his definition, only distributively generated near-rings were considered and the distributive generators played a vital role in the construction. In 1989, Le Riche, Meldrum and van der Walt [17], adopted a similar approach to that which led to a successful and fruitful definition of matrix near-rings, and defined the group near-ring in a more general sense. In particular, they defined R[G], the group near-ring of a group G over a near-ring R, as a subnear-ring of M(RG), the near-ring of all mappings of the group RG into itself. More recently, Groenewald and Lee [14], further generalised the definition of R[G] to R[S : M], the generalised semigroup near-ring of a semigroup S over any faithful R-module M. Again, the natural thing to do would be to extend the results obtained for R[G] to R[S : M], and this they achieved with much success.