Around Prüfer Extensions of Rings

Gabriel Picavet () and Martine Picavet-L’Hermitte
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Gabriel Picavet: Mathématiques
Martine Picavet-L’Hermitte: Mathématiques

A chapter in Algebraic, Number Theoretic, and Topological Aspects of Ring Theory, 2023, pp 351-382 from Springer

Abstract: Abstract This chapter intends to apply the properties of Prüfer extensions, investigated in the Knebusch–Zhang book, to ring extensions R ⊆ S. The integral closure R ¯ $$\overline R$$ of R in S is shown to be the intersection of all T ∈ [R, S], such that T ⊆ S is Prüfer. We are then able to establish an avoidance lemma for integrally closed subextensions. Rings of sections of the affine scheme defined by R provide results on S-regular ideals. Some results on pullbacks characterizations of Prüfer extensions are given. We introduce locally strong divisors, examining the properties of strong divisors of a local ring and their links with Prüfer extensions. The locally strong divisors allow us to give characterizations of QR extensions. We then derive some results on minimal and FCP extensions. Finally, we study the set of all primitive elements in an extension.

Keywords: Integral closure; Prüfer extension; Pullback; QR extension; Strong divisor; Ring of sections; Primary 13B02, 13B22, 13B40; Secondary 13B30 (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-28847-0_19

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DOI: 10.1007/978-3-031-28847-0_19

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