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Stability of ideals and its applications. (English) Zbl 1073.13508

Anderson, Daniel D. (ed.) et al., Ideal theoretic methods in commutative algebra. Proceedings of the conference in honor of Professor James A. Huckaba’s retirement, University of Missouri, Columbia, MO, USA. New York, NY: Marcel Dekker (ISBN 0-8247-0553-X/hbk). Lect. Notes Pure Appl. Math. 220, 319-341 (2001).
From the introduction: The notion, if not the terminology, of a stable ideal, an ideal that is projective over its ring of endomorphisms, is a familiar concept in the study of one-dimensional Noetherian rings, where it finds application in geometric contexts such as those involving Gorenstein singularities and Arf rings. A secondary aim of the present article is to survey some of the different manifestations of stable ideals in studies of Noetherian rings. After briefly reviewing the Noetherian case in section 2, we turn to the general case, the primary focus of the survey.
A commutative ring is stable if every regular ideal is stable. Stability is a much weaker requirement than what is demanded of Dedekind domains. For example, stable domains need not be Noetherian, integrally closed, coherent, nor one-dimensional. In fact, any cardinality can occur as the Krull dimension of a stable domain (see sections 3 and 4).
Beginning with a 1987 note by D. D. Anderson, J. Huckaba and I. Papick [Houston J. Math. 13, 13–17 (1987; Zbl 0624.13002)], some surprisingly definitive results on stability have been obtained for arbitrary integral domains. [See also the author’s papers: B. Olberding, J. Algebra 243, No. 1, 177–197 (2001; Zbl 1042.13013) and Commun. Algebra 30, No. 2, 877–895 (2002; Zbl 1073.13016)]. In this paper, we survey these results and report on some new ones. In a section of concluding remarks, we remark on some open questions.
For the entire collection see [Zbl 0964.00058].

MSC:

13G05 Integral domains
13C05 Structure, classification theorems for modules and ideals in commutative rings
13C10 Projective and free modules and ideals in commutative rings
13E15 Commutative rings and modules of finite generation or presentation; number of generators
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