Zero-Dimensionality and Products of Commutative Rings<P>

Robert Gilmer<P>

If $R$ is a Noetherian ring and $n$ is a positive integer, then there
are only finitely many ideals $I$ of $R$ such that the residue class
ring $R/I$ has cardinality $\leq n$.  If $R$ has Noetherian spectrum,
then the preceding statement holds for prime ideals of $R$.  Motivated
by this, we consider the dimension of an infinite product of
zero-dimensional commutative rings.  Such a product must be either
zero-dimensional or infinite-dimensional.  We consider the structure
of rings for which each subring is zero-dimensional and properties of
rings that are directed union of Artinian subrings.  Necessary and
sufficient conditions are given in order that an infinite product of
zero-dimensional rings be a directed union of Artinian subrings.