Ring coproducts embedded in power-series rings

Abstract

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 $α$ denote the natural map $A \amalg_R B \to R<< a,b>>$. This article describes some situations where $α$ 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 $α$ 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 $α$ is injective if $R$ is \textit{$Π$-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 $α$ 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 $α' \colon A \amalg_R B \to R<< a>>\amalg_R R<< b>>$ is not injective, and hence neither is $α$. 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 $α"\colon R<< a>>\amalg_R R<< b>> \to R<< a,b>>$ is not injective. Bergman gives procedures for constructing other examples where $α"$ is not injective.