Ring coproducts embedded in power-series rings
-
Pere Ara
and
Warren Dicks
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 ⨿RB → 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)±] ⨿RR[(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 ⨿RB → R〈〈a〉〉 ⨿RR〈〈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〉〉 ⨿RR〈〈b〉〉 → R〈〈a,b〉〉 is not injective. Bergman gives procedures for constructing other examples where α'' is not injective.
Funding source: DGI
Award Identifier / Grant number: MICIIN MTM2011-28992-C02-01
Funding source: Comissionat per Universitats i Recerca de la Generalitat de Catalunya
Funding source: Spain's Ministerio de Ciencia e Innovación
Award Identifier / Grant number: MTM2011-25955
We are very grateful to George Bergman for generously contributing the examples of Section 6 and suggesting many improvements of various earlier versions, to Dolors Herbera for providing much useful information for Section 4, and to the referee for the speed with which the article was assessed and the thought that went into the report.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Nielsen numbers of maps of aspherical figure-eight type polyhedra
- Simplicial geometry of unital lattice-ordered abelian groups
- Upper triangular matrices and operations in odd primary connective K-theory
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- Orthogonality of polynomials and orthosymmetry
- The moduli space of 4-dimensional non-nilpotent complex associative algebras
- Improvement of eigenfunction estimates on manifolds of nonpositive curvature
- Unramified representations of the universal central extension of SL2(𝔽q((z)))
- Languages associated with saturated formations of groups
- On some new model category structures from old, on the same underlying category
- On functions of bounded Λ-variation and integral smoothness
- Ring coproducts embedded in power-series rings
- A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces
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- The convergence of the product γ˜p-1h0bn-1 in the Adams spectral sequence
- Product graphs, sum-product graphs and sum-product estimates over finite rings
- BV functions in a Gelfand triple for differentiable measure and its applications
- The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn
- The Lefschetz coincidence class of p maps
- The natural pseudo-distance as a quotient pseudo-metric, and applications
- On a class of parametric Neumann problems with indefinite and unbounded potential
- An estimate on the heat kernel of Schrödinger operators with non-negative potentials on nilpotent Lie groups and its applications
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- Corrigendum to On some multiplicity and mixed multiplicity formulas [Forum Math. 26 (2014), 413–442]
Articles in the same Issue
- Frontmatter
- Nielsen numbers of maps of aspherical figure-eight type polyhedra
- Simplicial geometry of unital lattice-ordered abelian groups
- Upper triangular matrices and operations in odd primary connective K-theory
- Solving quintic equations by two-fold origami
- Orthogonality of polynomials and orthosymmetry
- The moduli space of 4-dimensional non-nilpotent complex associative algebras
- Improvement of eigenfunction estimates on manifolds of nonpositive curvature
- Unramified representations of the universal central extension of SL2(𝔽q((z)))
- Languages associated with saturated formations of groups
- On some new model category structures from old, on the same underlying category
- On functions of bounded Λ-variation and integral smoothness
- Ring coproducts embedded in power-series rings
- A uniform bound for geodesic periods of eigenfunctions on hyperbolic surfaces
- Action-induced nested classes of groups and jump (co)homology
- The convergence of the product γ˜p-1h0bn-1 in the Adams spectral sequence
- Product graphs, sum-product graphs and sum-product estimates over finite rings
- BV functions in a Gelfand triple for differentiable measure and its applications
- The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn
- The Lefschetz coincidence class of p maps
- The natural pseudo-distance as a quotient pseudo-metric, and applications
- On a class of parametric Neumann problems with indefinite and unbounded potential
- An estimate on the heat kernel of Schrödinger operators with non-negative potentials on nilpotent Lie groups and its applications
- A hybrid Euler–Hadamard product and moments of ζ'(ρ)
- Legendrian realization in convex Lefschetz fibrations and convex stabilizations
- Recollements induced from tilting modules over tame hereditary algebras
- Corrigendum to On some multiplicity and mixed multiplicity formulas [Forum Math. 26 (2014), 413–442]
