Ring coproducts embedded in power-series rings

TL;DR

This paper investigates conditions under which the natural map from the coproduct of subrings into a power-series ring is injective, providing new proofs of known results and identifying cases where injectivity fails.

Contribution

It offers new proofs for injectivity of coproduct embeddings in power-series rings and characterizes when such maps are not injective, extending previous results.

Findings

Injectivity holds when subrings are Ore localizations of polynomial rings.

The map is injective if the ring is extit{ extbf{$oldsymbol{ ext{Π}}$-semihereditary}}.

Counterexamples are provided for rings with higher global dimension, showing non-injectivity.

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 $\alpha$ 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.