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Finitely Stable Rings

In: Commutative Algebra

Author

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  • Bruce Olberding

    (New Mexico State University, Department of Mathematical Sciences)

Abstract

A commutative ring R is finitely stable provided every finitely generated regular ideal of R is projective as a module over its ring of endomorphisms. This class of rings includes the Prüfer rings, as well as the one-dimensional local Cohen-Macaulay rings of multiplicity at most 2. Building on work of Rush, we show that R is finitely stable if and only if its integral closure R ¯ $$\overline{R}$$ is a Prüfer ring, every R-submodule of R ¯ $$\overline{R}$$ containing R is a ring and every regular maximal ideal of R has at most 2 maximal ideals in R ¯ $$\overline{R}$$ lying over it. This characterization is deduced from a more general theorem regarding what, motivated by work of Knebusch and Zhang, we term a finitely stable subring R of a ring between R and its complete ring of quotients.

Suggested Citation

  • Bruce Olberding, 2014. "Finitely Stable Rings," Springer Books, in: Marco Fontana & Sophie Frisch & Sarah Glaz (ed.), Commutative Algebra, edition 127, pages 269-291, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4939-0925-4_16
    DOI: 10.1007/978-1-4939-0925-4_16
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