Ring coproducts embedded in power-series rings

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.