Products of commutative rings and zero-dimensionality

Abstract

If R is a Noetherian ring and is a positive integer, then there are only finitely many ideals I of R such that the residue class ring R/I has cardinality < 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.