Homomophic images of an infinite product of zero-dimensional rings

Abstract

Let R=∏α∊A be an infinite product of zero-dimensionalchained rings. It is known that R is either zero-dimensional or infinitedimensional. We prove that a finite-dimensional homo~norphic image of R is of dimension at most one. If each R, is a PIR and if R is infinite-dimensional, then R admits one-dimensional hornomorphic images. However, without the PIR hypothesis on the rings Rα, we present examples to show that R may be infinite-dimensional while each finite-dimensional homomorphic image of R is zero-dimensicnal. JVe 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 honlomorphic image.