Tame and wild primes in direct products of commutative rings

TL;DR

This paper classifies prime ideals in infinite direct products of commutative rings into tame and wild types, revealing their structural properties, connections to ultrafilters, and implications for the prime spectrum's topology.

Contribution

It introduces the concepts of tame and wild primes, characterizes their properties, and links wild primes to ultrafilters, advancing understanding of prime spectra in infinite product rings.

Findings

Tame primes form an open subscheme of the prime spectrum.

Wild primes contain the direct sum ideal and are uncountably many.

The prime spectrum has two types of connected components: tame and wild.

Abstract

A complete understanding of the structure of all prime ideals of an infinite direct product of commutative rings (e.g. in terms of more specific objects) has remained a challenging problem for decades. In this article, new advances have been made in this regard. We observe that in an infinite direct product of nonzero rings there are two different types of prime ideals, that we call tame primes and wild primes. Among the main results, we prove that the set of tame primes is an open subscheme of the prime spectrum, and this scheme is non-affine if and only if the index set is infinite. As an application, a prime ideal is a wild prime if and only if it contains the direct sum ideal. Next, we show that an uncountable number of (wild) primes of an infinite direct product ring are induced by the (non-principal) ultrafilters of the index set (at least $2^{\mathfrak{c}}$ wild primes, $\mathfrak{c}$ is the cardinality of the continuum). This description has an important consequence: if a direct product ring has a wild prime, then the set of wild primes is infinite (uncountable). The connected components of the prime spectrum of an infinite direct product ring are also investigated. We observe that, like for prime ideals, there are two types of connected components in this space, tame ones and wild ones.