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
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.