Homomorphic Images of an Infinite Product of Zero-dimensional Rings

R. Gilmer, W. Heinzer

Let R = \prod_{a \in A}R_a be an infinite product of zero-dimensional chained rings. It is known that R is either zero-dimensional or infinite-dimensional. We prove that a finite-dimensional homomorphic image of R is of dimension at most one. If each R_a is a PIR and if R is infinite-dimensional, then R admits one-dimensional homomorphic images. However, without the PIR hypothesis on the rings R_a, we present examples to show that R may be infinite-dimensional while each finite-dimensional homomorphic image of R is zero-dimensional. We prove that a prime ideal of R of positive height is of infinite height, and we give conditions for an infinite product of zero-dimensional local rings to admit a one-dimensional local domain as a homomorphic image.