Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Axiomatics of Free Group Rings   In [FGRS1,FGRS2] the relationship between the universal and elementary theory
of a group ring $R[G]$ and the corresponding universal and elementary theory of
the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is
a commutative ring with identity $1 \ne 0$. Of course, these are relative to an
appropriate logical language $L_0,L_1,L_2$ for groups, rings and group rings
respectively. Axiom systems for these were provided in [FGRS1]. In [FGRS1] it
was proved that if $R[G]$ is elementarily equivalent to $S[H]$ with respect to
$L_{2}$, then simultaneously the group $G$ is elementarily equivalent to the
group $H$ with respect to $L_{0}$, and the ring $R$ is elementarily equivalent
to the ring $S$ with respect to $L_{1}$. We then let $F$ be a rank $2$ free
group and $\mathbb{Z}$ be the ring of integers. Examining the universal theory
of the free group ring ${\mathbb Z}[F]$ the hazy conjecture was made that the
universal sentences true in ${\mathbb Z}[F]$ are precisely the universal
sentences true in $F$ modified appropriately for group ring theory and the
converse that the universal sentences true in $F$ are the universal sentences
true in ${\mathbb Z}[F]$ modified appropriately for group theory. In this paper
we show this conjecture to be true in terms of axiom systems for ${\mathbb
Z}[F]$.

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The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org
9
The categories of this paper are: nlin
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The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
6
The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: solv-int
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The categories of this paper are: bayes-an