Intermediate ideals in matrix near-rings

One of the ways in which matrix near-rings as defined by Meldrum and Ms 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.