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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 ℤ  p  , the free pro-Hall  S  bin -group ( A , S  bin ) is fully residually free pro- p . Furthermore, we prove that any finite set of elements in ( A , S  bin ) defines a pro- p  subgroup and so an irreducible coordinate group over the free pro- p  group.

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