On the ubiquity of Arf rings

TL;DR

This paper introduces the concept of weakly Arf rings, generalizing Arf rings, and explores their properties, characterizations, and examples in various algebraic constructions.

Contribution

It develops the theory of weakly Arf rings, providing characterizations and examples, and clarifies their relationship with Arf rings and ring closedness.

Findings

Weakly Arf rings are characterized and distinguished from Arf rings.

Examples include idealizations, fiber products, and invariant subrings.

The relationship between weakly Arf rings and strict closedness is established.

Abstract

We introduce and develop the theory of weakly Arf rings, which is a generalization of Arf rings, initially defined by J. Lipman in 1971. We provide characterizations of weakly Arf rings and study the relation between these rings, the Arf rings, and the strict closedness of rings. Furthermore, we give various examples of weakly Arf rings that come from idealizations, fiber products, determinantal rings, and invariant subrings.