Pro-Hall $R$-groups and groups discriminated by the free pro-$p$ group

TL;DR

This paper introduces pro-Hall R-groups as inverse limits of Hall R-groups and demonstrates that certain free pro-Hall groups over rings discriminated by _p are fully residually free pro-_p, with applications to coordinate groups.

Contribution

It defines pro-Hall R-groups and proves that free pro-Hall groups over rings discriminated by _p are fully residually free pro-_p, linking algebraic structures and residual properties.

Findings

Free pro-Hall S^{bin}-groups are fully residually free pro-_p.

Finite sets in these groups define pro-_p subgroups.

These groups serve as irreducible coordinate groups over free pro-_p.

Abstract

In this note we introduce pro-Hall $R$-groups as inverse limits of Hall $R$-groups and show that for the binomial closure $S^{bin}$ of any ring $S$ discriminated by $\mathbb{Z}_p$, the free pro-Hall $S^{bin}$-group $\mathbb{F}(A,S^{bin})$ is fully residually free pro-$p$. Furthermore, we prove that any finite set of elements in $\mathbb{F}(A,S^{bin})$ defines a pro-$p$ subgroup and so an irreducible coordinate group over the free pro-$p$ group.