Intermediate ideals in matrix near-rings

Abstract

One of the ways in which matrix near-rings as defined by Meldrum and van der Walt differ from matrix rings is that the correspondence between ideals in the matrix near-ring and those in the base near-ring is many-one, unlike the ring case. Here it is shown that, in suitable cases, an arbitrarily large lattice of ideals in the matrix near-ring can be made to correspond to an ideal in the base near-ring. This is achieved in two quite different contexts, using different properties of near-rings. The relationship between the ideals in the matrix near-ring and those in the base near-ring is also examined in some detail. Matrix near-rings as defined by Meldrum and van der Walt [7] have proved to be a fruitful concept. One of the ways in which they behave differently from matrix rings is that the two most obvious ways of extending an ideal from the base near-ring to the matrix near-ring do not always yield the same object, unlike the situation with rings. This leads to the idea of an intermediate ideal, one that lies between the two extensions of the ideal in the base near-ring. We show here that such ideals exist in abundance. This is done in two very different contexts. The first context is that of polynomial near-rings, whose addition is commutative, and where it is the multiplicative structure which is the key to the construction. The second context is that of d. g. weakly distributive near-rings, where the additive structure plays an important role and the multiplicative structure is very simple. We also point out some elementary properties of intermediate ideals before starting on the details of the constructions. But first an introduction to the whole subject.