Pere Ara and Warren Dicks (2012)

# Ring coproducts embedded in power-series rings

arXiv:1211.6323.

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)ˆ\pm 1\] amalg\_R R[ (1+b)ˆ\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 aamalg\_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 aamalg\_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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