## 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*〉〉. Let α denote the natural
map *A* ⨿_{R} *B* → *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*)^{±}] ⨿_{R} *R*[(1+*b*)^{±}],
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
Π-*semihereditary*, that is, every finitely generated,
torsionless, right *R*-module is projective. (This concept was first
studied by M. F. Jones, who showed that it is left-right
symmetric. It follows from a result of
I. I. Sahaev that
if w.gl.dim *R* ≤ 1 and *R* embeds in a skew field,
then *R* is Π-semihereditary. Also, it follows from a result of
V. C. Cateforis
that if *R* is right semihereditary and right self-injective,
then *R* is Π-semihereditary.)

The arguments and results extend easily from two variables to any set of variables.

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
w.gl.dim *R* ≥ 2,
then there exist examples where the map
α': *A* ⨿_{R} *B* → *R*〈〈*a*〉〉 ⨿_{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, then the map
α'': *R*〈〈*a*〉〉 ⨿_{R} *R*〈〈*b*〉〉
→ *R*〈〈*a*,*b*〉〉 is not injective. Bergman gives procedures for constructing
other examples where α'' is not injective.

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