Chains of intermediate ideals in matrix near-rings

Abstract

0. Introduction. It is well-known that there is a one to one correspondence between the set of(two-sided) ideals of a ring R with identity and the set of ideals of the corresponding matrix ring ~ (R), for any n e N. Moreover, the correspondence appears in a very trivial and natural way, as the reader would have seen in many a textbook. Since the introduction of matrices over arbi trary near-rings by Meldrum and van der Walt ([4]) in /984, ideals in the near-ring R and related ideals in the corresponding matrix near-ring JM~ (R) have been extensively studied. In 1987 van der Walt ([7]) has shown that there are in general more ideals in a matrix near-ring than in its base near-ring. He did this by relating ideals in the matrix near-ring to a given ideal in the base near-ring in two natural ways. These two methods coincide when the base near-ring is a ring, but not in the general near-ring setting, as an example in [7] shows. This was the first evidence of the preponderance of ideals in 1M~, (R) over those in R. Since then, more examples of this kind were constructed, even for near-rings which are "very nearly" rings. (See [t] and [3]). In a recent investigation by Meldrum and the author (see [2]), this process is t aken a step further: we have shown that, apart from the additional ideal in ]M, (R) that one can expect by performing the two natural constructions discussed in the previous paragraph on an ideal in R, there are, in general, even more ideals the so-called intermediate ideals lurking around in a matrix near-ring. One of the examples given in [2] was a zero-symmetric abelian near-ring and it was conjectured that arbitrarily long chains of intermediate ideals can be constructed by using this example. In this note it is shown that such chains of intermediate ideals do indeed exist.