The representation of d. g. near-rings

Abstract

In this paper we study the problem of the representation of d.g. near-rings, and in particular the problem of a faithful representation, which is equivalent to the adjoining of an identity. This problem has been considered by Malone [5] and Malone and Heatherly [6] and [7]. They have shown that a finite near-ring with two sided zero can be embedded in the d.g. near-ring generated by the inner automorphisms of a suitable group, and that an identity can always be adjoined to a near-ring with two sided zero. They have also given some special conditions under which a faithful representation of a d.g. near-ring exists.From another point of view, Frohlich has studied groups over a d.g. nearring in [3] and [4J. If {R, S) is a d.g. near-ring, where S is the distributive semigroup generating R, then he showed that free (R, S) groups exist. We use free (R,S) groups to show that not every d.g. near-ring (R, S) can have a faithful representation on a group, if we insist that S should be a semigroup of distributive elements, i.e. endomorphisms on the group. This is true even in the finite case.We start by setting the work of Frohlich on free (R. S) groups in the context of varieties, using methods differing substantially from his. Using these ideas, we construct in each non-abelian variety a d.g. near-ring without a faithful representation. This opens up the problem of determining those d.g. near-rings which do have a faithful representation. It also leaves open the question of whether it is possible to embed a d.g. near-ring {R, S) in a d.g. near-ring with identity, if we do not insist that the elements of S should be distributive in the larger near-ring.We finish by establishing that for every d.g. near-ring (R,S), there exist 'nearest' d.g. near-rings (R,S), (R,S) which have faithful representations and such that (R,S) is a homomorphic image of (R,S) and (R,S) is a homomorphic image of (R, S). For those d.g. near-rings to which an identity can be adjoined, there is a natural way of doing it. If the near-ring is a ring, then it is interesting to note that this method of adjoining the identity is the standard one. 467 468 • J. D. P. Meldrum [2] Definitions and preliminary results[3]The representation of d.g. near rings 469

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Presentations of faithful d.g. near-rings

“…For any d.g. near-ring, the free (R, S) group on a set X exists and is denoted ¥r{X, R, S) (Section 2 of Meldrum (2)). If X = {x} is a one element set, we write Fr( In this paper, we consider d.g.…”

“…We-remark that we could restrict ourselves to working within a variety "V of additive groups as was done in Meldrum (2), and all the results will go over in such a context. Now assume that the result is true for m = q and let the length of D(U') = q + 1.…”

“…We follow these for the most part, except that we use left near-rings rather than right near-rings. We follow exactly an earlier paper, Meldrum (2), where there are detailed definitions and many results relating to faithful d.g. near-rings.…”