  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.
