Infinite products of zero-dimensional commutative rings

Robert Gilmer, William Heinzer

Suppose $\{R_a\}_{a \in A}$ is a family of zero-dimensional principal ideal rings (PIR's) and let $R = \prod_{a \in A} R_a$. If $R$ is of positive dimension, it is known that:

(*) If $I$ is the direct sum ideal of $R$ and $J \subseteq I$ is an ideal of $R$, then each extension ring of $R/J$ is infinite-dimensional, or equivalently, $R/J$ cannot be imbedded in a finite-dimensional ring. In particular, each extension ring of $R$ is infinite-dimensional.

(**) If $S$ is a nonzero homomorphic image of $R$, then $S$ has dimension either 0, 1, or $\infty$, and all three of these values are, in fact, realized as the dimension of a homomorphic image of $R$.

In this paper we consider these assertions in the case where $R = \prod_{a \in A} R_a$, where $\{R_a\}_{a \in A}$ is a family of zero-dimensional commutative rings. We prove that statement (*) holds more generally for a product of zero-dimensional rings---that is, if $\{R_a\}_{a \in A}$ is a family of zero-dimensional rings and $R = \prod_{a \in A} R_a$ is infinite-dimensional, then each extension ring of $R$ or of $R/J$, where $J$ is contained in the direct sum ideal $I = \bigoplus_{a \in A}R_a$, is infinite-dimensional.

The situation with regard to (**), however, is different. In fact we exhibit, for each positive integer $d$, a product $R$ of zero-dimensional local rings such that the values realizable as the dimension of a homomorphic image of $R$ are precisely $0, 1, 2, \dots d, \infty$. We also establish existence of a product of zero-dimensional local rings for which there exists a homomorphic image of dimension $d$ for each positive integer $d$.