Polygons in Quadratically Closed Rings and Properties of n-adically Closed Rings

TL;DR

This paper extends properties of absolutely integrally closed rings to a broader class called n-adically closed rings, providing elementary proofs and exploring their structural properties and intersection behaviors.

Contribution

It introduces n-adically closed rings, proves they lack polygons, and characterizes the irreducible intersection property within this class.

Findings

Absolutely integrally closed rings have no polygons.

n-adically closed rings generalize quadratically closed rings.

The paper provides necessary and sufficient conditions for the irreducible intersection property.

Abstract

This paper is inspired by Michael Artin's paper "On The Join of Hensel Rings". In his paper, Artin proves that in an absolutely integrally closed ring the sum of two prime ideals is either prime or the whole ring. A more elementary proof of the previous statement is given in M. Hochster and C. Huneke's "Infinite Integral Extensions and Big Cohen-Macaulay Algebras" for the larger class of quadratically closed rings. Motivated by Hochster and Huneke's proof, we give an elementary proof of the fact that an absolutely integrally closed ring has no polygons. We prove this for the larger class of 2n-adically closed rings, after defining n-adically closed rings. The notion of irreducible intersection property is introduced, and a necessary and sufficient condition is proved for a ring to have this property. Other ring and scheme theoretic properties of absolutely integrally closed rings and n-adically closed rings are shown in detail.