New methods toward the patch and flat topologies with applications
aa r X i v : . [ m a t h . A C ] F e b ON THE FINITENESS OF MINIMAL AND MAXIMALSPECTRA
ABOLFAZL TARIZADEH
Abstract.
In this article, it is proved that the minimal spec-trum of a commutative ring is quasi-compact with respect to theflat topology. Consequently, all of the related results of Kaplan-sky, Anderson, Gilmer-Heinzer and Bahmanpour et al on minimalprimes are deduced as special cases of this result. Dually, somesimilar results are also obtained for maximal ideals. Introduction
Recently in [9] we have rediscovered the Hochster’s inverse topology(see [3, Prop. 8]) on the prime spectrum by a new and purely algebraicmethod. We call it the flat topology (it is worthy to mention thatduring the writing [9] we were not aware of Hochster’s work). Hencethe flat topology and Hochster’s inverse topology are exactly the samethings. Roughly speaking, for a given ring R , then the collection ofsubsets V( I ) = { p ∈ Spec( R ) : I ⊆ p } where I runs through the set off.g. ideals of R formes a basis for the opens of the flat topology, see [9,Theorem 3.2]. We use f.g. in place of “finitely generated”.Certainly Hochster’s work is seminal. But some aspects of the in-verse topology are hidden in the voluminous of Hochster’s work. Forinstance, there are some major results in the literature on the finite-ness of minimal primes which have been proved independently whilemany of these results are easily implied from the basic properties ofthe inverse topology. One of the main aims of this article is to showthe simplicity of some of these results. Indeed, we obtain Theorem 2.1which is the first main result of this note. All of the results [7, Theorem88], [5], [6, Theorem 1.6] and [4, Theorem 2.1] are special cases and canbe easily deduced from this Theorem. The machinery of deep resultsof commutative algebra are applied to prove the final main result ofthis note, Theorem 2.7. All of the rings which are considered in this note are commutative. 2. Main results
Theorem 2.1.
Let R be a ring and consider the flat topology over Spec( R ) . Then Min( R ) the minimal spectrum of R is quasi-compact. Proof.
Suppose Min( R ) ⊆ S α U α where for each α , U α is a flat opensubset of Spec( R ). We claim that Spec( R ) = S α U α . Let p be a primeideal of R . There is a minimal prime p ′ of R such that p ′ ⊆ p . Thereexists some α such that p ′ ∈ U α . We have p ∈ U α . If not, then { p } ⊆ U cα = Spec( R ) \ U α . By [9, Corollary 3.6], { p } = { q ∈ Spec( R ) : q ⊆ p } .This is a contradiction. This establishes the claim. By [9, Remark 3.5],the flat topology is quasi-compact. (cid:3) The main result of [4, Theorem 2.1] is a special case of Theorem 2.1:
Corollary 2.2.
Let R be a ring and consider the flat topology over Spec( R ) . Then the following conditions are equivalent. (i) The set
Min( R ) is finite. (ii) For every minimal prime p there is an element f ∈ p such that { p } = Min( R ) ∩ V( f ) . (iii) Every minimal prime is an open point of the subspace
Min( R ) . Proof. (i) ⇒ (ii) : By the Prime avoidance theorem [1, Tag 00DS],there is an element f ∈ p such that f / ∈ S p ′ ∈ Min( R ) , p ′ = p p ′ . (ii) ⇒ (iii) : There is nothing to prove. (iii) ⇒ (i) : It is an immediate consequence of Theorem 2.1. (cid:3) Theorem 2.1 vastly generalizes [7, Theorem 88], [5] and [6, Theorem1.6]:
Corollary 2.3.
Let R be a ring such that every minimal prime is theradical of a f.g. ideal. Then R has a finitely many minimal primes. First proof.
It implies from Theorem 2.1 or Corollary 2.2.
N THE FINITENESS OF MINIMAL AND MAXIMAL SPECTRA 3
Second proof.
Here, we present another proof for it without usingthe previous results. Let p be a minimal prime of R . By the hypothe-sis, there is a f.g. ideal I of R such that V( p ) = V( I ). Therefore V( p )is an open subset of Spec( R ) w.r.t. the patch (constructible) topology.We have Spec( R ) = S p ∈ Min( R ) V( p ). The patch topology is compact, see[9, Proposition 2.4]. Therefore Min( R ) is a finite set. (cid:3) The converse of Corollary 2.3 does not necessarily hold, see [4, Ex-ample 2.14].
Remark 2.4.
Let { R i : i ∈ I } be a family of domains. For each i , π − i (0) is a minimal prime of R = Q i ∈ I R i where π i : R → R i is thecanonical projection. Because, suppose there exists a prime ideal p of R such that p ⊂ π − i (0). Pick a ∈ π − i (0) \ p . Then ab = 0 where b = ( δ i,j ) j ∈ I . It follows that b ∈ p , a contradiction. The minimal prime π − i (0) is principal since it is generated by the sequence (1 − δ i,j ) j ∈ I .Using this and Corollary 2.3 then we find a minimal prime p in everyinfinite direct product ring Q Z such that p = π − i (0) for all i . More-over, p is not the radical of a f.g. ideal. It is worthy to mention thatthe set of minimal primes of Q Z is in bijective correspondence to theset of all ultrafilters on the index set, see [8, Proposition 1].As a dual of Theorem 2.1, we have: Proposition 2.5.
Let R be a ring and consider the Zariski topologyover Spec( R ) . Then Max( R ) the maximal spectrum of R is quasi-compact. Proof.
It is well-known. Indeed, it is proved exactly like Theorem2.1. (cid:3)
Corollary 2.6.
Let R be a ring and consider the Zariski topology over Spec( R ) . Then the following conditions are equivalent. (i) The set
Max( R ) is finite. (ii) For each maximal ideal m , m + T m ′ ∈ Max( R ) , m ′ = m m ′ = R . (iii) For each maximal ideal m there is an element f ∈ R \ m such that { m } = Max( R ) ∩ D ( f ) . ABOLFAZL TARIZADEH (iv)
Each maximal ideal is an open point of the subspace
Max( R ) . Proof.
All of the implications (i) ⇒ (ii) , (ii) ⇒ (iii) and (iii) ⇒ (iv) are obvious. The implication (iv) ⇒ (i) implies from Proposition2.5. (cid:3) As a dual of Corollary 2.3, we have the following non-trivial result.
Theorem 2.7.
Let R be a ring with the property that for each maximalideal m the canonical map π : R → R m is injective and of finite type.Then R has a finitely many maximal ideals. Proof.
By [2, Theorem 1.1], π is of finite presentation. Everyflat ring homomorphism which is also of finite presentation then itinduces a Zariski open map between the corresponding prime spectra,see [1, Tag 00I1]. Therefore there are finite subsets { f , ..., f n } ⊆ R and { m , ..., m n } ⊆ Max( R ) such that R = ( f , ..., f n ) and D ( f i ) = Im π ∗ i for all i where π i : R → R m i is the canonical map. It follows thatMax( R ) = { m , ..., m n } . (cid:3) References [1] Aise Johan de Jong et al, Stacks Project, see http://stacks.math.columbia.edu.[2] Cox. Jr. S, Rush. D, Finiteness in flat modules and algebras, Journal of Alge-bra, Volume 32, Issue 1, 1974, p. 44-50.[3] Melvin. Hochster, Prime ideal structure in commutative rings, Trans. Amer.Math. Soc. 142 (1969), 43-60.[4] Kamal. Bahmanpour et al, A note on minimal prime divisors of an ideal,Algebra Colloquium. 18 (2011) 727-732.[5] D. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math. Soc.,122, No. 1. (1994), 13-14.[6] R. Gilmer, W. Heinzer, Primary ideals with finitely generated radical in acommutative ring, Manuscripta Math. 78 (1993) 201-221.[7] Irving. Kaplansky, Commutative rings, revised edition, Univ. of Chicago Press,Chicago, 1974.[8] Ronnie. Levy et al, The prime spectrum of an infinite product of copies of Z ,Fundamenta Mathematicae. 138 (1991) 155-164.[9] Abolfazl. Tarizadeh, Flat topology and its dual aspects, submitted,arXiv:1503.04299v9 [math.AC]. Department of Mathematics, Faculty of Basic Sciences, Universityof Maragheh, P. O. Box 55136-553, Maragheh, Iran.
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