Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Polygons in Quadratically Closed Rings and Properties of n-adically
  Closed Rings   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.

19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph