Chapter

Commutative Algebra

pp 269-291

Date:

Finitely Stable Rings

  • Bruce OlberdingAffiliated withDepartment of Mathematical Sciences, New Mexico State University Email author 

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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 \(\overline{R}\) is a Prüfer ring, every R-submodule of \(\overline{R}\) containing R is a ring and every regular maximal ideal of R has at most 2 maximal ideals in \(\overline{R}\) 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.

Keywords

Stable ideal Finitely stable ring Prüfer ring Prüfer extension

Mathematics Subject Classification (2011):

13F05 13B22 13C10