Absolute integral closures of commutative rings

TL;DR

This paper investigates the properties and existence of absolute integral closures in commutative rings, revealing uniqueness conditions and constructing universal closures for specific ring classes.

Contribution

It characterizes when absolute integral closures are unique and constructs universal closures for certain subrings, advancing understanding of their structure in commutative algebra.

Findings

All rings have absolute integral closures, but they are not always unique.

Finite products of domains are the only reduced rings with finitely many minimal primes where closures are unique.

Universal absolute integral closures exist for certain subrings of products of domains.

Abstract

Absolute integral closures of general commutative unital rings are explored. All rings admit absolute integral closures, but in general they are not unique. Among the reduced rings with finitely many minimal prime ideals, finite products of domains are the only rings for which they are unique. Arguments using model theory suggest that the same holds for all infinite rings that are finite products of connected rings. Universal absolute integral closures, which contain every aic of a given ring, are shown to exist for certain subrings of products of domains.