Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local dimension theory of tensor products of algebras over a ring   Our main goal in this paper is to set the general frame for studying the
dimension theory of tensor products of algebras over an arbitrary ring $R$.
Actually, we translate the theory initiated by A. Grothendieck and R. Sharp and
subsequently developed by A. Wadsworth on Krull dimension of tensor products of
algebras over a field $k$ into the general setting of algebras over an
arbitrary ring $R$. For this sake, we introduce and study the notion of a
fibred AF-ring over a ring $R$. This concept extends naturally the notion of
AF-ring over a field introduced by A. Wadsworth in \cite{W} to algebras over
arbitrary rings. We prove that Wadsworth theorems express local properties
related to the fibre rings of tensor products of algebras over a ring. Also,
given a triplet of rings $(R,A,B)$ consisting of two $R$-algebras $A$ and $B$
such that $A\otimes_RB\neq \{0\}$, we introduce the inherent notion to
$(R,A,B)$ of a $B$-fibred AF-ring which allows to compute the Krull dimension
of all fiber rings of the considered tensor product $A\otimes_RB$. As an
application, we provide a formula for the Krull dimension of $A\otimes_RB$ when
$A$ and $B$ are $R$-algebras with $A$ is zero-dimensional as well as for the
Krull dimension of $A\otimes_{\mathbb{Z}}B$ when $A$ is a fibred AF-ring over
the ring of integers $\mathbb{Z}$ with nonzero characteristic and $B$ is an
arbitrary ring. This enables us to answer a question of Jorge Matinez on
evaluating the Krull dimension of $A\otimes_{\mathbb{Z}}B$ when $A$ is a
Boolean ring. Actually, we prove that if $A$ and $B$ are rings such that
$A\otimes_{\mathbb{Z}}B$ is not trivial and $A$ is a Boolean ring, then
dim$(A\otimes_{\mathbb{Z}}B)=\mbox {dim}\Big (\displaystyle {\frac B{2B}}\Big
)$.

18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: supr-con
36
The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
1
The categories of this paper are: cs
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The categories of this paper are: quant-ph