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The use of ultrafilters to study the structure of Prüfer and Prüfer-like rings

In: Commutative Algebra

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  • K. Alan Loper

    (Ohio State University-Newark, Department of Mathematics)

Abstract

The notion of a sequence of objects converging to a limit point is very natural in the context of topology. In a collection of articles from the last couple decades ultrafilters were employed to great effect in forcing sequences of prime ideals or valuation domains to converge without reference to topology. The context of these results was the study of Prüfer domains and rings of integer-valued polynomials. More recently Fontana and Loper clarified these results by demonstrating that ultrafilters can be used to define a topology which is equivalent to the classical constructible or patch topology. The history of these results is recounted in this expository article in chronological sequence.

Suggested Citation

  • K. Alan Loper, 2011. "The use of ultrafilters to study the structure of Prüfer and Prüfer-like rings," Springer Books, in: Marco Fontana & Salah-Eddine Kabbaj & Bruce Olberding & Irena Swanson (ed.), Commutative Algebra, pages 319-333, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-6990-3_12
    DOI: 10.1007/978-1-4419-6990-3_12
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