Avoidance and absorbance
Introduction
The prime avoidance lemma is one of the most fundamental results in commutative algebra: if an ideal is contained in a finite union of prime ideals, then it is already contained in one of them. The set-theoretic dual result – referred to as prime absorbance – is also useful (and follows directly from the definition of primeness): if a finite intersection of ideals is contained in a prime ideal, then one of them is already contained in the prime. However, both results fail for infinite families in general. For example, infinite prime avoidance already fails in the ring (cf. Example 2.1(5)), and infinite prime absorbance fails in the ring of integers .
With this in mind, the main goal of this paper is to study the dual notions of prime avoidance and prime absorbance, especially in the infinite case. Infinite prime avoidance has been periodically investigated over the years, see e.g. [2], [10], [12], [14], and [15]. Dually, infinite prime absorbance has been studied in [11, §V] and [16, §4]. In [6], the prime avoidance lemma is also investigated for non-commutative rings.
Contrary to what one might initially expect, avoidance is not strictly limited to prime ideals. In Section 2, we formulate the avoidance property in general and show that it passes to intersections in a certain specific sense, cf. Theorem 2.5. This allows us to generalize the classical prime avoidance lemma to radical ideals, cf. Theorem 2.2, Theorem 2.3.
Section 3 investigates the rings in which every set of primes has the avoidance property, the so-called compactly packed (or C.P.) rings. Dually, Section 4 investigates the rings in which every set of primes has the absorbance property, which we name a properly zipped (or P.Z.) ring. Although they have received less attention in the literature, P.Z. rings admit a number of interesting and natural characterizations, e.g. a ring is P.Z. if and only if any union of Zariski-closed sets is Zariski-closed, for further characterizations see Theorem 4.2.
A recurring theme is the interplay of the C.P. and P.Z. properties with chain conditions and Noetherian-like properties. For instance, it is shown that P.Z. rings are semilocal, and satisfy d.c.c. on both prime ideals and finitely generated radical ideals, cf. Corollary 4.3, Corollary 4.4. Theorem 4.8 characterizes the rings which are both C.P. and P.Z. as the rings with only finitely many prime ideals. The flat topology on spectra of C.P. and P.Z. rings is also investigated, see Proposition 3.5, Proposition 4.5. The P.Z. rings of dimension 1 are characterized in Theorem 4.10. Finally, Section 5 concludes with various examples.
In this paper, all rings are commutative with . The nilradical is denoted by . If f is a member of a ring R, then and . There is a (unique) topology, called the flat topology, on for which the collection of , where I is a finitely generated ideal of R, forms a base of open sets. If is a prime ideal of R, then is the flat closure of the point . For more information see e.g. [16].
Section snippets
General avoidance
Let R be a ring, and a set of ideals of R. We say that satisfies avoidance if for any ideal J of R, whenever , then for some .
Example 2.1 We illustrate the avoidance property with some basic examples: Any set of 2 ideals satisfies avoidance: if an ideal is contained in a union of 2 ideals, then it is contained in one of them. Any finite set of prime ideals satisfies avoidance: this is the classical prime avoidance lemma. If a ring R contains an infinite field k, then any finite set of ideals
Prime avoidance and C.P. rings
We now investigate the rings in which every set of primes satisfies avoidance – these are the so-called compactly packed (or C.P.) rings. That is, R is C.P. if whenever an ideal I is contained in the union of a family of prime ideals, then for some i. C.P. rings have been studied in the literature, see e.g. [10], [12] and [15].
We first record various ring-theoretic constructions which preserve the C.P. property:
Proposition 3.1 Let R be a ring. If is C.P., then R is C.P. If R is C.P., then so is any
Prime absorbance and P.Z. rings
The dual notion of a C.P. ring can be defined as follows. We say that a ring R is a properly zipped (or P.Z.) ring if whenever a prime ideal of R contains the intersection of a family of prime ideals of R, then for some i.
We first give the analogue of Proposition 3.1, whose proof we leave as an exercise:
Proposition 4.1 Let R be a ring. If is P.Z., then R is P.Z. If R is P.Z., then so is any quotient or localization of R. A finite product of P.Z. rings is P.Z.
Next, we turn towards a
Examples
Example 5.1 If k is a field, then is a Noetherian ring of dimension 1, but is neither C.P. nor P.Z.: for more details see [2, Example 4].
Example 5.2 If is an infinite family of rings, then always has infinitely many minimal primes and maximal ideals. By Remark 3.4(1) and Corollary 4.4, is never C.P. nor P.Z. Thus Propositions 3.1(3) and 4.1(3) are sharp.
Example 5.3 In Theorems 3.3(vi) and 4.2(vii), the “radical” assumptions are necessary. For example, if R is a DVR with a uniformizer p, then R is
Acknowledgements
We would like to give thanks to the referee for very careful reading of the paper and for his/her valuable comments which improved the paper.
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