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A prime ideal principle in commutative algebra. (English) Zbl 1168.13002
Let \(R\) be a commutative ring with identity. It is a “metatheorem” in commutative algebra that an ideal maximal with respect to some property is often prime. Of course the best known and probably most important is Krull’s result that an ideal maximal with respect to missing a multiplicatively closed set is prime. Also, an ideal maximal with respect to not being principal, invertible, or finitely generated or an ideal maximal among annihilators of nonzero elements of a module is prime. This delightful paper actually gives such a metatheorem, the Prime Ideal Principle.
Let \(\mathfrak F\) be a family of ideals of \(R\) with \(R\in\mathfrak F\). Then \(\mathfrak F\) is an Oka family (resp., Ako family) if for an ideal \(I\) of \(R\) and \(a, b\in R\), \((I, a), (I : a)\in \mathfrak F\) implies \(I\in \mathfrak F\) (resp., \((I, a), (I, b)\) implies \((I, ab)\in\mathfrak F\)). The Prime Ideal Principle states that if \(\mathfrak F\) is an Oka or Ako family, then the complement of the family \(\mathfrak F^c\subseteq\text{Spec}(R)\). Hence if \(\mathfrak F\) is an Oka or Ako family in \(R\) and every nonempty chain of ideals in \(\mathfrak F\) has an upper bound in \(\mathfrak F\) and all primes belong to \(\mathfrak F\), then all ideals of \(R\) belong to \(\mathfrak F\).
From these two results we recapture the results listed above plus many more and the well known consequences such as \(R\) is noetherian (resp. a Dedekind domain, a PIR) if every nonzero prime ideal is finitely generated (resp., invertible, principal). The paper studies Oka and Ako families and related types of families in detail.
Many more applications of the Prime Ideal Principal are given, some of them new such as the following: a ring \(R\) is Artinian if and only if for each prime ideal \(P\) of \(R\), \(P\) is finitely generated and \(R/P\) is finitely cogenerated. The work is also interpreted in terms of categories of cyclic modules.
This paper was a joy to read and should be read by all those interested in commutative algebra.

13A15 Ideals and multiplicative ideal theory in commutative rings
prime ideal
Full Text: DOI
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