Homomorphic Images of an Infinite Product of Zero-dimensional Rings<P>

R. Gilmer, W. Heinzer<P>

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.