aa r X i v : . [ m a t h . A C ] J un AVOIDANCE AND ABSORBANCE
ABOLFAZL TARIZADEH AND JUSTIN CHEN
Abstract.
We study the two dual notions of prime avoidanceand prime absorbance. We generalize the classical prime avoidancelemma to radical ideals. A number of new criteria are provided foran abstract ring to be C.P. (every set of primes satisfies avoid-ance) or P.Z. (every set of primes satisfies absorbance). Specialconsideration is given to the interaction with chain conditions andNoetherian-like properties. It is shown that a ring is both C.P. andP.Z. iff it has finite spectrum. Introduction
The prime avoidance lemma is one of the most fundamental resultsin commutative algebra: if an ideal is contained in a finite union ofprime 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 in-tersection of ideals is contained in a prime ideal, then one of them isalready contained in the prime. However, both results fail for infinitefamilies in general. For example, infinite prime avoidance already failsin the ring k [ x, y ] (cf. Example 2.1(5)), and infinite prime absorbancefails in the ring of integers Z .With this in mind, the main goal of this paper is to study the dualnotions of prime avoidance and prime absorbance, especially in the in-finite case. Infinite prime avoidance has been periodically investigatedover the years, see e.g. [1], [8], [10], [12], and [13]. Dually, infiniteprime absorbance has been studied in [9, § V] and [14, § Mathematics Subject Classification.
Key words and phrases.
Prime avoidance, Prime absorbance, C.P. rings.
Section 3 investigates the rings in which every set of primes hasthe avoidance property, the so-called compactly packed (or C.P.) rings.Dually, Section 4 investigates the rings in which every set of primes hasthe 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. iff any union of Zariski-closed sets is Zariski-closed.A recurring theme is the interplay of the C.P. and P.Z. propertieswith chain conditions and Noetherian-like properties. For instance, itis shown that P.Z. rings are semilocal, and satisfy d.c.c. on both primeideals and finitely generated radical ideals. Theorem 4.8 characterizesthe rings which are both C.P. and P.Z. as the rings with only finitelymany prime ideals. The flat topology on spectra of C.P. and P.Z.rings is also investigated, see Propositions 3.5 and 4.5. The P.Z. ringsof dimension 1 are characterized in Theorem 4.10. Finally, Section 5concludes with various examples.In this paper, all rings are commutative with 1 = 0. The nilradicalis denoted by N . For a ring R , there is a (unique) topology, called theflat topology, on Spec( R ) for which the collection of V ( I ), where I is afinitely generated ideal of R , forms a base of open sets. If p is a primeideal of R , then Λ( p ) := { q ∈ Spec( R ) : q ⊆ p } is the flat closure ofthe point p ∈ Spec( R ). For more information see e.g. [14].2. General avoidance
Let R be a ring, and S a set of ideals of R . We say that S satisfiesavoidance if for any ideal J of R , whenever J ⊆ [ I ∈ S I , then J ⊆ I forsome I ∈ S . Example 2.1.
We illustrate the avoidance property with some basicexamples:(1) Any set of ≤ R contains an infinite field k , then any finite set of ideals satisfiesavoidance: no k -vector space is a finite union of proper subspaces.(4) The set of maximal ideals Max( R ) satisfies avoidance: if an idealconsists of nonunits, then it is contained in a maximal ideal.(5) The avoidance property need not pass to subsets or supersets. Forinstance a maximal ideal may be contained in the union of theother maximal ideals (e.g. ( x, y ) in k [ x, y ]). VOIDANCE AND ABSORBANCE 3
In the following result, the prime avoidance lemma is generalized forradical ideals.
Theorem 2.2. ( Radical avoidance ) If an ideal I of a ring R is con-tained in the union of a finite family { I k } of radical ideals of R , then I ⊆ I k for some k .Proof. Suppose I is not contained in any of the I k , so there exists f k ∈ I \ I k for each k . Then I k does not meet the multiplicative set { , f k , f k , . . . } , so there is a prime ideal p k of R containing I k that alsodoes not contain f k . Then clearly I ⊆ S k p k , but this is in contradictionwith the prime avoidance lemma. (cid:3) Remark 2.3.
In Theorem 2.2, just like in the usual prime avoidancelemma [11, Theorem 3.61], we may assume that two of the I k ’s arearbitrary ideals (not necessarily radical).Similarly, the version of prime avoidance given by Edward Davis (cf.e.g. [6, Ex. 16.8] or [11, Theorem 3.64]) can also be generalized toradical ideals. Corollary 2.4. ( Davis’ radical avoidance ) Let { I k } be a finite familyof radical ideals of a ring R and f ∈ R . If I is an ideal of R such that Rf + I * S k I k , then there exists g ∈ I such that f + g / ∈ S k I k .Proof. For each k , there exists a prime ideal p k of R such that I k ⊆ p k but Rf + I * p k . So by prime avoidance, Rf + I * S k p k . By Davis’prime avoidance, there exists g ∈ I such that f + g / ∈ p k for all k . (cid:3) The above results exemplify the fact that some properties of primeideals can be generalized to radical ideals. In fact, what the proof ofTheorem 2.2 shows is that the avoidance property passes to intersec-tions in the following sense:
Theorem 2.5.
Let S be a set of ideals of R , and let T be the set ofall intersections of ideals in S . If every subset of S satisfies avoidance,then every subset of T satisfies avoidance as well.Proof. Let { I t | t ∈ T } be a subset of T , and suppose J ⊆ [ t ∈ T I t , but J I t for all t ∈ T . For every t ∈ T , choose f t ∈ J \ I t . Since I t isan intersection of elements of S , there exists K t ∈ S such that I t ⊆ K t and f t K t . Then J ⊆ [ t ∈ T I t ⊆ [ t ∈ T K t , but J K t for all t , so { K t | t ∈ T } ⊆ S does not satisfy avoidance, a contradiction. (cid:3) A. TARIZADEH AND J. CHEN
Note that if S = Spec R , then the proof of Theorem 2.5, with classicalprime avoidance, yields an alternative proof of Theorem 2.2.3. Prime avoidance and C.P. rings
We now investigate the rings in which every set of primes satisfiesavoidance – 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 ofa family { p i } of prime ideals, then I ⊆ p i for some i . C.P. rings havebeen studied in the literature, see e.g. [8], [10] and [13].We first record various ring-theoretic constructions which preservethe C.P. property: Proposition 3.1.
Let R be a ring. (1) If R/ N is C.P., then R is C.P. (2) If R is C.P., then so is any quotient or localization of R . (3) A finite product of C.P. rings is C.P.Proof.
We illustrate the proof of (1). Assume R/ N is C.P. If { p i } areprimes of R and I is an ideal of R with I ⊆ S i p i , then ( I + N ) / N ⊆ S i p i / N . Then ( I + N ) / N ⊆ p i / N for some i , so I ⊆ p i .The proofs of (2) and (3) are straightforward and left as exercises. (cid:3) We now turn towards various characterizations of the C.P. property.As we will see, the C.P. property turns out to imply Noetheriannessof the prime spectrum in the Zariski topology, so we first recall somecriteria for this to occur.
Proposition 3.2.
Consider the following conditions on a ring R : ( i ) Spec( R ) is a Noetherian space in the Zariski topology ( ii ) R satisfies the ascending chain condition on radical ideals ( iii ) Every radical ideal of R is the radical of a finitely generatedideal ( iv ) Every ideal of R has only finitely many minimal primes.Then ( i ) ⇔ ( ii ) ⇔ ( iii ) ⇒ ( iv ) , and ( iv ) ⇒ ( i ) if dim R is finite.Proof. Cf. [7, Propositions 1.1 and 2.1]. (cid:3)
We are now ready to state our characterizations of C.P. rings. Thefollowing result improves on [8, Theorem 1], [10, Theorem 1.1] and [13]with the addition of conditions ( v ) and ( vi ). Theorem 3.3.
The following are equivalent for a ring R : ( i ) R is a C.P. ring. VOIDANCE AND ABSORBANCE 5 ( ii ) If a prime ideal p of R is contained in the union of a family { p i } of prime ideals of R , then p ⊆ p i for some i . ( iii ) Every radical ideal of R is the radical of a principal ideal. ( iv ) Every prime ideal of R is the radical of a principal ideal. ( v ) Spec( R ) is a Noetherian space in the Zariski topology, and forany f, g ∈ R there exists h ∈ R such that V ( f ) ∩ V ( g ) = V ( h ) . ( vi ) If an ideal I of R is contained in the union of a family { I k } ofradical ideals of R , then I ⊆ I k for some k .Proof. ( i ) ⇔ ( ii ) is an easy exercise, see e.g. [13].For ( i ) ⇒ ( iii ) ⇒ ( iv ) ⇒ ( i ) see [13] or [8, Theorem 1].( iii ) ⇒ ( v ) : By Proposition 3.2( iii ), Spec( R ) is Noetherian. For any f, g ∈ R , by hypothesis there exists h ∈ R such that p ( f, g ) = p ( h ).It follows that V ( f ) ∩ V ( g ) = V ( f, g ) = V ( h ).( v ) ⇒ ( iii ) : Let I be a radical ideal of R . By Proposition 3.2( iii ),there exists a finitely generated ideal J of R such that V ( I ) = V ( J ).By hypothesis, there exists h ∈ R such that V ( J ) = V ( h ). It followsthat I = p ( h ).( ii ) ⇔ ( vi ) : This follows immediately from Theorem 2.5. (cid:3) Remark 3.4. (1) It follows from Proposition 3.2 and Theorem 3.3that every C.P. ring satisfies the ascending chain condition on rad-ical ideals and has finitely many minimal primes.(2) Recall that the arithmetic rank of an ideal I is the least number ofelements required to generate I up to radical, i.e.ara I := inf (cid:8) n | ∃ a , . . . , a n ∈ R with p ( a , . . . , a n ) = √ I (cid:9) Another way to phrase the proof of Theorem 3.3 is: ( v ) says exactlythat ara I < ∞ for all ideals I and if ara I < ∞ , then ara I ≤ I ≤ I , which is ( iii ).(3) Since ht I ≤ ara I for all ideals in a Noetherian ring, it follows fromTheorem 3.3( iv ) that a Noetherian C.P. ring has dimension ≤ Proposition 3.5.
Let R be a C.P. ring. (1) The collection of V ( f ) with f ∈ R forms a base for the flatopens of Spec( R ) . (2) The flat closed subsets of
Spec( R ) are precisely of the form Im π ∗ where π : R → S − R is the canonical map and S is amultiplicative subset of R .Proof. (1): If I is a finitely generated ideal of R , then by Theorem3.3( iii ) there exists f ∈ R such that √ I = p ( f ), so V ( I ) = V ( f ). A. TARIZADEH AND J. CHEN (2): Clearly every subset of the given form is flat closed. Conversely,if E ⊆ Spec R is a flat closed then E ⊆ Im π ∗ where S := R \ S p ∈ E p and π : R → S − R is the canonical map. If q ∈ Im π ∗ then q ⊆ S p ∈ E p . Since R is C.P., this implies q ⊆ p for some p ∈ E . It follows that q ∈ E ,since each flat closed subset is stable under generalization. Therefore E = Im π ∗ . (cid:3) Prime absorbance and P.Z. rings
The dual notion of a C.P. ring can be defined as follows. We say thata ring R is a properly zipped (or P.Z.) ring if whenever a prime ideal p of R contains the intersection of a family { p i } of prime ideals of R ,then p i ⊆ p for some i .We first give the analogue of Proposition 3.1, whose proof we leaveas an exercise: Proposition 4.1.
Let R be a ring. (1) If R/ N is P.Z., then R is P.Z. (2) If R is P.Z., then so is any quotient or localization of R . (3) A finite product of P.Z. rings is P.Z.
Next, we turn towards a characterization of P.Z. rings. The followingresult simplifies and improves on [14, Theorem 4.2], with the additionof condition ( vii ). Theorem 4.2.
The following are equivalent for a ring R : ( i ) R is a P.Z. ring. ( ii ) If p is a prime ideal of R , then there exists f ∈ R such that Λ( p ) = D ( f ) . ( iii ) If p is a prime ideal of R , then there exists f ∈ R \ p such thatthe canonical map R f → R p is an isomorphism. ( iv ) If p is a prime ideal of R , then the localization map R → R p isof finite presentation. ( v ) The Zariski opens of
Spec( R ) are stable under arbitrary inter-sections. ( vi ) Spec( R ) is a Noetherian space in the flat topology. ( vii ) If a prime ideal p of R contains the intersection of a family { I k } of radical ideals of R , then I k ⊆ p for some k .Proof. ( i ) ⇒ ( ii ) : If X := Spec( R ) \ Λ( p ), then I := T q ∈ X q is notcontained in p . Thus there exists f ∈ I \ p . It follows that Λ( p ) = D ( f ).( ii ) ⇒ ( iii ) : If g ∈ R \ p then g/ R f , since Λ( p ) = D ( f ). Thus by the universal property of localization, there exists a VOIDANCE AND ABSORBANCE 7 (unique) ring map ϕ : R p → R f such that π f = ϕ ◦ π p where π f : R f → R p and π p : R → R p are the canonical maps. It follows that ϕ ◦ π f and π f ◦ ϕ are the identity maps.( iii ) ⇒ ( iv ) : The isomorphism R f ∼ = R [ X ] / ( f X −
1) gives that R → R p is of finite presentation.( iv ) ⇒ ( ii ) : It is well known that every flat ring map which is alsoof finite presentation induces a Zariski open map on prime spectra.In particular, Λ( p ) is a Zariski open of Spec( R ). Thus we may writeΛ( p ) = S i D ( f i ). Therefore p ∈ D ( f k ) for some k , and so Λ( p ) = D ( f k ).( ii ) ⇒ ( v ) : It suffices to prove the assertion for basic Zariski opens.If p ∈ T i D ( f i ) then there exists f ∈ R such that Λ( p ) = D ( f ). Itfollows that p ∈ D ( f ) ⊆ T i D ( f i ).( v ) ⇒ ( vi ) : It suffices to show that every flat open U of Spec( R )is quasi-compact. If U = S k V ( I k ) where each I k is a finitely generatedideal of R , then by hypothesis, U = V ( I ) for some ideal I of R . Butfor any ring R , V ( I ) is a quasi-compact subset of Spec( R ) in the flattopology.( vi ) ⇒ ( ii ) : Since Λ( p ) is a flat closed subset of Spec( R ), thereexists a finitely generated ideal I of R such that Spec( R ) \ Λ( p ) = V ( I ),because every subspace of a Noetherian space is quasi-compact. ThusΛ( p ) = D ( f ) for some f ∈ I .( ii ) ⇒ ( vii ) : Choose f ∈ R such that Λ( p ) = D ( f ). Then f / ∈ I k for some k , so I k ∩ S = ∅ where S = { , f, f , . . . } . Hence there existsa prime ideal q of R such that I k ⊆ q and f / ∈ q . Thus I k ⊆ q ⊆ p .( vii ) ⇒ ( i ) : Clear. (cid:3) The next two corollaries codify important properties of P.Z. rings,and serve as a dual to Remark 3.4(1).
Corollary 4.3.
Every P.Z. ring satisfies the descending chain condi-tion on both prime ideals and finitely generated radical ideals.Proof.
Let R be a P.Z. ring. If p ⊇ p ⊇ . . . is a descending chain ofprime ideals of R , then Λ( p ) ⊇ Λ( p ) ⊇ . . . is a descending chain offlat closed subsets of Spec( R ). By Theorem 4.2( vi ), there is some n such that p n = p i for all i > n .Similarly, if I ⊇ I ⊇ . . . is a descending chain of finitely generatedradical ideals of R , then V ( I ) ⊆ V ( I ) ⊆ . . . is an ascending chain offlat open subsets of Spec( R ). Again by Theorem 4.2( vi ), there is some n such that I n = I k for all k > n . (cid:3) Corollary 4.4.
Let R be a P.Z. ring. Then Max( R ) is a finite set. A. TARIZADEH AND J. CHEN
Proof.
By Theorem 4.2( ii ), for every maximal ideal m of R , there exists x m ∈ R such that Λ( m ) = D ( x m ). Since Max( R ) is always Zariskiquasi-compact, there is a finite subcover Max( R ) ⊆ d S i =1 D ( x i ) where x i := x m i for i = 1 , . . . , d . Then Max( R ) = { m , . . . , m d } . (cid:3) Proposition 4.5.
For a P.Z. ring R , the flat opens of Spec( R ) areprecisely sets of the form V ( I ) with I a finitely generated ideal of R .Proof. Clearly every subset of the given form is flat open. Conversely,let U be a flat open subset of Spec( R ). Then by Theorem 4.2( vi ), U is quasi-compact in the flat topology. Thus there exists a finitelygenerated ideal I of R such that U = V ( I ). (cid:3) We next characterize the zero-dimensional C.P. and P.Z. rings:
Theorem 4.6.
The following are equivalent for a ring R : ( i ) R is a zero-dimensional C.P. ring. ( ii ) R is a zero-dimensional P.Z. ring. ( iii ) R/ N is a finite product of fields.Proof. Suppose dim R = 0. Then Min( R ) = Max( R ) = Spec( R ), soif R is C.P. (resp. P.Z.), then Spec( R ) is finite by Remark 3.4(1)(resp. Corollary 4.4), say Spec( R ) = { p , . . . , p n } . Thus by the Chineseremainder theorem, R/ N ∼ = R/ p × . . . × R/ p n is a finite product offields. This shows ( i ) ⇒ ( iii ) (resp. ( ii ) ⇒ ( iii )).On the other hand, a field is both C.P. and P.Z., so the conversefollows from Propositions 3.1 and 4.1. (cid:3) Corollary 4.7.
Let R be a Boolean ring. Then R is C.P. if and onlyif R is P.Z. if and only if R is finite. In particular, for a set X , thepower set ring P ( X ) is C.P. or P.Z. if and only if X is finite.Proof. By Theorem 4.6 it suffices to show that a Boolean P.Z. ring isfinite. This follows from the proof of Theorem 4.6( ii ) ⇒ ( iii ), sinceevery residue field of a Boolean ring is Z / Z . (cid:3) We can now characterize the rings which satisfy both avoidance andabsorbance. The proof relies significantly on topological methods.
Theorem 4.8.
The following are equivalent for a ring R : ( i ) R is both C.P. and P.Z. ( ii ) Spec( R ) is Noetherian in both the Zariski and flat topologies. ( iii ) Spec( R ) is a finite set.Proof. ( i ) ⇒ ( ii ) : See Theorem 3.3( v ) and Theorem 4.2( vi ).( ii ) ⇒ ( iii ) : See [14, Theorem 4.5]. VOIDANCE AND ABSORBANCE 9 ( iii ) ⇒ ( i ) : Clear. (cid:3) Returning to P.Z. rings, the following proposition states that if thering itself is Noetherian (not just the spectrum), then it suffices to checkonly minimal primes in Theorem 4.2( ii ), i.e. a Noetherian ring is P.Z.iff every minimal prime is an isolated point in the Zariski topology.This is a dual result to [8, Theorem 1] which may be phrased as: aNoetherian ring is C.P. iff every maximal ideal is an isolated point inthe flat topology. Theorem 4.9.
Let R be a Noetherian ring. Then R is P.Z. if andonly if for every minimal prime p of R , there exists f ∈ R such that D ( f ) = { p } .Proof. The direction ⇒ follows from Theorem 4.2( ii ). Conversely, itsuffices to show that V ( p ) is finite for all p ∈ Min( R ) (so that Spec( R )is finite). Pick p ∈ Min( R ). If V ( p ) is infinite then it contains infinitelymany height one prime ideals, all of which contain f . Then the imageof f in R/ p generates a height one ideal with infinitely many minimalprimes, which is impossible since R/ p is Noetherian. (cid:3) We next rephrase the P.Z. property for rings of dimension 1:
Theorem 4.10.
Let R be a 1-dimensional ring. Then R is P.Z. iff (1) R is semilocal, (2) For all p ∈ Min( R ) , \ q ∈ Min( R ) \{ p } q = N , and (3) For all m ∈ Max( R ) , m + \ q ∈ Min( R ) \ Λ( m ) q = R .Proof. Any P.Z. ring satisfies conditions (2) and (3), as well as (1) byCorollary 4.4. Conversely, we show that conditions (1)-(3) are equiv-alent to Theorem 4.2( ii ). If m ∈ Max( R ), then by (3) we may choose f m ∈ (cid:16) \ q ∈ Min( R ) \ Λ( m ) q (cid:17) \ m . Then choosing g m ∈ (cid:16) \ m ′ ∈ Max( R ) \{ m } m ′ (cid:17) \ m (which is possible since the intersection is finite) gives Λ( m ) = D ( f m g m ). If now p is a non-maximal (hence minimal) prime, then(2) implies that there exists x p ∈ (cid:16) \ q ∈ Min( R ) \{ p } q (cid:17) \ p . As before, if y p ∈ (cid:16) \ m ∈ V ( p ) ∩ Max( R ) m (cid:17) \ p , then Λ( p ) = { p } = D ( x p y p ). (cid:3) Remark 4.11.
Any ring with finitely many minimal primes satisfiesconditions (2) and (3) in Theorem 4.10.
Corollary 4.12.
Let R be a 1-dimensional reduced local ring. Then R is P.Z. iff \ q ∈ Min( R ) \{ p } q = 0 for all p ∈ Min( R ) .Proof. This follows immediately from Theorem 4.10. (cid:3)
Finally, we investigate Goldman ideals in P.Z. rings. The followingresult improves on [3, Theorems 18, 24] and [9, § I, Prop. 1] with theaddition of conditions ( vi ) and ( vii ). Theorem 4.13.
The following are equivalent for an integral domain R with field of fractions K : ( i ) K is a simple extension of R . ( ii ) K is a finitely generated algebra over R . ( iii ) There exists a maximal ideal m of R [ X ] such that m ∩ R = 0 . ( iv ) { } is a locally closed subset of Spec( R ) . ( v ) The zero ideal of R is an isolated point of Spec( R ) . ( vi ) If { p i } is a family of non-zero prime ideals of R , then T i p i = 0 . ( vii ) There is some non-zero f ∈ R such that for each non-zero g ∈ R then f ∈ p ( g ) .Proof. For ( i ) ⇔ ( ii ) ⇔ ( iii ) see [3, Theorems 18, 24]. For ( iii ) ⇔ ( iv ) ⇔ ( v ) see [9, § I, Prop. 1].( v ) ⇒ ( vi ) : It suffices to show that the intersection of all non-zeroprimes is non-zero. By hypothesis, there is some non-zero f ∈ R suchthat { } = D ( f ). Thus f belongs to the above intersection.( vi ) ⇒ ( vii ) : Choose some non-zero f from the intersection of allnon-zero primes. Then f ∈ p ( g ) for all non-zero g .( vii ) ⇒ ( i ) : If h/g ∈ K then g is non-zero. Thus there exists anatural number n > f n = rg for some non-zero r ∈ R . So h/g = rh/f n , which yields K = R [1 /f ]. (cid:3) An integral domain which satisfies one of the equivalent conditionsof Theorem 4.13 is called a Goldman domain. A prime ideal p of aring R is called a Goldman ideal (or G -ideal) of R if R/ p is a Goldmandomain. The set of Goldman ideals of a ring R is denoted by Gold( R ).For more information on this set see [9]. Corollary 4.14.
If a ring R is a P.Z. ring, then Gold( R ) = Spec( R ) .Proof. Let p be a prime ideal of R . If ( p i ) is a subset of V ( p ) \ { p } then T i p i = p . Thus by Theorem 4.13( vi ), p is a G -ideal. (cid:3) VOIDANCE AND ABSORBANCE 11 Examples
Example 5.1. If k is a field, then k [ x, y, z ] / ( xy, xz ) is a Noetherianring of dimension 1, but is neither C.P. nor P.Z.: for more details see[1, Example 5]. Example 5.2. If { R i } is an infinite family of rings, then Q i R i alwayshas infinitely many minimal primes and maximal ideals. By Remark3.4(1) and Corollary 4.4, Q i R i is never C.P. nor P.Z. Thus Propositions3.1(3) and 4.1(3) are sharp. Example 5.3.
In Theorems 3.3( vi ) and 4.2( vii ), the “radical” assump-tions are necessary. For example, if R is a DVR with a uniformizer p ,then R is P.Z., but T k ≥ ( p k ) = 0. Similarly, if R is a Dedekind do-main with torsion but nonzero class group, then R is C.P., but for anynonprincipal ideal I , one has I ⊆ S a ∈ I ( a ), but I ( a ) for any a ∈ I . Example 5.4.
Neither C.P. nor P.Z. are local properties: if R is aninfinite product of fields (or more generally any non-Noetherian ab-solutely flat ring), then R is neither C.P. nor P.Z., by Example 5.2.However, R p is a field for all p ∈ Spec( R ).Similarly, neither the C.P. nor P.Z. properties are preserved by ad-joining variables. For example if k is a field then k [ x ] is not P.Z., sinceMax( k [ x ]) is infinite. Also k [ x ] is C.P. (being a PID), but by Remark3.4(3), k [ x ][ y ] = k [ x, y ] is not C.P.We have seen that among Noetherian rings, the C.P. or P.Z. ringshave dimension ≤
1. However, in general there are rings of any fi-nite dimension which are both C.P. and P.Z. For example, a finite-dimensional valuation ring has finite spectrum (since the primes aretotally ordered), hence is both C.P. and P.Z. More generally, it is wellknown that for any finite poset P , there exists a ring R such thatSpec( R ) is order-isomorphic to P as a poset, see [5] or [2].The following result guarantees existence of P.Z. rings with infinitespectra. Proposition 5.5.
Let R be a ring such that Spec( R ) is a Noetherianspace in the Zariski topology. Then there exists a P.Z. ring R ′ such that Spec( R ′ ) is in bijection with Spec( R ) and this correspondence reversesthe prime orders.Proof. There exists a ring R ′ such that Spec( R ′ ) equipped with theflat topology is homeomorphic to Spec( R ) equipped with the Zariskitopology, see [2, Theorem 6 and Proposition 8] or [14, Theorem 3.20]. Thus Spec( R ′ ) is a Noetherian space in the flat topology, so by Theorem4.2( vi ), R ′ is a P.Z. ring. If the homeomorphism is denoted by ∗ :Spec( R ) → Spec( R ′ ), then for primes p ⊆ q of R , one has q ∈ V ( p ) = { p } . Thus q ∗ ∈ { p ∗ } = Λ( p ∗ ), so q ∗ ⊆ p ∗ . (cid:3) Example 5.6.
Taking R = Z in Proposition 5.5 yields a 1-dimensionallocal P.Z. ring R ′ with infinite spectrum. Since Corollary 4.4 impliesthat a P.Z. ring with infinite spectrum has Krull dimension >
0, this isa minimal example of a P.Z. ring with infinite spectrum. Quotientingby the nilradical of R ′ then gives a P.Z. ring satisfying the conditionsof Corollary 4.12. References [1] J. Chen, Infinite prime avoidance, arXiv:1710.05496v1.[2] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer Math.Soc. 142 (1969) 43-60.[3] I. Kaplansky, Commutative rings, The University of Chicago Press (1974).[4] O. A. Karamzadeh, The prime avoidance lemma revisited, Kyungpoook Math.J. 52 (2012) 149-153.[5] W. J. Lewis, The spectrum of a ring as a partially ordered set, J. Algebra,25(3) (1973) 419-434.[6] H. Matsumura, Commutative ring theory, Cambridge University Press (1989).[7] J. Ohm and R. L. Pendleton, Rings with Noetherian spectrum, Duke Math. J.35(3) (1968) 631-639.[8] J. V. Pakala and T. S. Shores, On compactly packed rings, Pacific J. Math.97(1) (1981) 197-201.[9] G. Picavet, Autor des id´eaux premiers de Goldman d’un aneau commutatif,Ann. Univ. Clermont-Ferrand 57(11) (1975) 73-90.[10] C. Reis and T. Viswanathan, A compactness property of prime ideals in Noe-therian rings, Proc. Amer Math. Soc. 25(2) (1970) 353-356.[11] R. Y. Sharp, Steps in commutative algebra, Cambridge University Press(2000).[12] R. Y. Sharp and P. V´amos, Baire’s category theorem and prime avoidance incomplete local rings, Arch. Math. 44 (1985) 243-248.[13] W. Smith, A covering condition for prime ideals, Proc. Amer. Math. Soc. 30(3)(1971) 451-452.[14] A. Tarizadeh, Flat topology and its dual aspects, Comm. Algebra 47(1) (2019)195-205.
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