Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On minimal ring extensions   Let $R$ be a commutative ring with identity. The ring $R\times R$ can be
viewed as an extension of $R$ via the diagonal map $\Delta: R \hookrightarrow
R\times R$, given by $\Delta(r) = (r, r)$ for all $r\in R$. It is shown that,
for any $a, b\in R$, the extension $\Delta(R)[(a,b)] \subset R\times R$ is a
minimal ring extension if and only if the ideal $<a-b>$ is a maximal ideal of
$R$. A complete classification of maximal subrings of $R(+)R$ is also given.
The minimal ring extension of a von Neumann regular ring $R$ is either a von
Neumann regular ring or the idealization $R(+)R/\mathfrak{m}$ where
$\mathfrak{m}\in \text{Max}(R)$. If $R\subset T$ is a minimal ring extension
and $T$ is an integral domain, then $(R:T) = 0$ if and only if $R$ is a field
and $T$ is a minimal field extension of $R$, or $R_J$ is a valuation ring of
altitude one and $T_{J}$ is its quotient field.

9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat