ON ZERO-DIMENSIONALITY AND FRAGMENTED RINGS

Abstract. A commutative ring R is said to be fragmented if each nonunit of R is divisible by all positive integral powers of some corresponding nonunit of R. It is shown that each fragmented ring which contains a nonunit non-zero-divisor has (Krull) dimension ∞. We consider the interplay between fragmented rings and both the atomic and the antimatter rings. After developing some results concerning idempotents and nilpotents in fragmented rings, along with some relevant examples, we use the “fragmented ” and “locally fragmented ” concepts to obtain new characterizations of zero-dimensional rings, von Neumann regular rings, finite products of fields, and fields