HOMOMORPHIC IMAGES OF AN INFINITE PRODUCT OF ZERO-DIMENSIONAL RINGS

Let R = PI(a epsilon A) R(a) be an infinite product of zero-dimensiona l chained rings. It is known that R is either zero-dimensional or infi nite-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 i nfinite-dimensional, then R admits one-dimensional homomorphic images. However, without the PIR hypothesis on the rings R(a), we present exa mples to show that R may be infinite-dimensional while each finite-dim ensional homomorphic image of R is zero-dimensional. We prove that a p rime 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.