Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rigid and Non-Rigid Mathematical Theories: the Ring $\mathbb{Z}$ Is
  Nearly Rigid   Mathematical theories are classified in two distinct classes : {\it rigid},
and on the other hand, {\it non-rigid} ones. Rigid theories, like group theory,
topology, category theory, etc., have a basic concept - given for instance by a
set of axioms - from which all the other concepts are defined in a unique way.
Non-rigid theories, like ring theory, certain general enough pseudo-topologies,
etc., have a number of their concepts defined in a more free or relatively
independent manner of one another, namely, with {\it compatibility} conditions
between them only. As an example, it is shown that the usual ring structure on
the integers $\mathbb{Z}$ is not rigid, however, it is nearly rigid.

15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga