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

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.