Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Ring coproducts embedded in power-series rings   Let $R$ be a ring (associative, with 1), and let $R<< a,b>>$ denote the
power-series $R$-ring in two non-commuting, $R$-centralizing variables, $a$ and
$b$. Let $A$ be an $R$-subring of $R<< a>>$ and $B$ be an $R$-subring of $R<<
b>>$, and let $\alpha$ denote the natural map $A \amalg_R B \to R<< a,b>>$.
This article describes some situations where $\alpha$ is injective and some
where it is not.
  We prove that if $A$ is a right Ore localization of $R[a]$ and $B$ is a right
Ore localization of $R[b]$, then $\alpha$ is injective. For example, the group
ring over $R$ of the free group on $\{1+a, 1+b\}$ is $R[ (1+a)^{\pm 1}]
\amalg_R R[ (1+b)^{\pm 1}]$, which then embeds in $R<< a,b>>$. We thus recover
a celebrated result of R H Fox, via a proof simpler than those previously
known.
  We show that $\alpha$ is injective if $R$ is \textit{$\Pi$-semihereditary},
that is, every finitely generated, torsionless, right $R$-module is projective.
  The article concludes with some results contributed by G M Bergman that
describe situations where $\alpha$ is not injective. He shows that if $R$ is
commutative and $\text{w.gl.dim\,} R \ge 2$, then there exist examples where
the map $\alpha' \colon A \amalg_R B \to R<< a>>\amalg_R R<< b>>$ is not
injective, and hence neither is $\alpha$. It follows from a result of K R
Goodearl that when $R$ is a commutative, countable, non-self-injective, von
Neumann regular ring, the map $\alpha"\colon R<< a>>\amalg_R R<< b>> \to R<<
a,b>>$ is not injective. Bergman gives procedures for constructing other
examples where $\alpha"$ is not injective.

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The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
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The categories of this paper are: physics
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The categories of this paper are: math-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: funct-an