Embedding of near-rings

Abstract

This dissertation provides an elementary proof that every near-ring can be embedded in a near-ring with identity and in fact in T(G), the near-ring of all transformations on some group G. Conditions for embedding a near-ring (R, +, [dot]) in T(R) are also considered. The ideal Δ(R) = {a ε R : xa = 0 for each x ε R} plays a key part in the investigation. These points are developed in Chapter III. Blackett has investigated the ideal structure of T(G) and T₀(G) for G finite. T₀(G) is the subnear-ring in of T(G) composed of all those mappings on G which carry the group zero into itself. Chapter II investigates T(G) and T₀(G) for an arbitrary group G. The groups T(G) and T₀(G) are direct sums of copies of the group (G, +). Letting α[superscript x, subscript y] ε T₀(G) be defined by (t)α[superscript x, subscript y] = {[top equation: 0 if t does not equal x, bottom equation: y if t = x,] and A[subscript x] = {α[superscript x, subscript y] : y ε G},for x [does not equal] 0, it is shown that each A[subscript x] is a minimal right ideal of T₀(G) generated by the idempotent and α[superscript x, subscript x] and A = ΣΦ A[subscript x], x ε G - {0} is a right ideal in T₀(G). A = T₀(G) if and only if G is finite. Maximal ideals in T₀(G) are also considered. In Chapter IV the work of Clay and Malone on near-rings defined on cyclic groups and simple groups is extended. It is shown that if (G, +) is a simple group and (G, +, [dot]) a near-ring with a non-zero right distributive element, then either ab = 0 for each a, b ε G or (G, +, [dot]) is a field. ...