IMBEDDABILITY OF A COMMUTATIVE RING IN A FINITE-DIMENSIONAL RING

In the category of commutative unitary rings we prove that, for each p ositive integer n, there exists an n-dimensional ring R(n) that is not a subring of a ring of dimension less than n. We also prove existence of a ring with a unique maximal ideal that is not a subring of a fini te-dimensional ring. A case of the imbeddability problem of particular interest is that for R = PI R(alpha), where each R(alpha) is zero-dim ensional. It is known that R is either zero-dimensional or infinite-di mensional. In the case where each R(alpha) is a PIR and R is infinite- dimensional, we show that neither R nor R/I, where I is the direct sum ideal of R, is imbeddable in a finite-dimensional ring; moreover, R a dmits a DVR as a homomorphic image.