Zero-Dimensionality and Products of Commutative Rings

Robert Gilmer

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.