arXiv:1609.00947v5 [] 22 Feb 2018

ON THE FINITENESS OF MINIMAL AND MAXIMAL
SPECTRA
ABOLFAZL TARIZADEH
Abstract. In this article, it is proved that the minimal spectrum of a commutative ring is quasi-compact with respect to the
flat topology. Consequently, all of the related results of Kaplansky, Anderson, Gilmer-Heinzer and Bahmanpour et al on minimal
primes are deduced as special cases of this result. Dually, some
similar results are also obtained for maximal ideals.

1. 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 algebraic
method. We call it the flat topology (it is worthy to mention that
during the writing [9] we were not aware of Hochster’s work). Hence
the flat topology and Hochster’s inverse topology are exactly the same
things. Roughly speaking, for a given ring R, then the collection of
subsets V(I) = {p ∈ Spec(R) : I ⊆ p} where I runs through the set of
f.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 inverse topology are hidden in the voluminous of Hochster’s work. For
instance, there are some major results in the literature on the finiteness of minimal primes which have been proved independently while
many of these results are easily implied from the basic properties of
the inverse topology. One of the main aims of this article is to show
the simplicity of some of these results. Indeed, we obtain Theorem 2.1
which is the first main result of this note. All of the results [7, Theorem
88], [5], [6, Theorem 1.6] and [4, Theorem 2.1] are special cases and can
be easily deduced from this Theorem. The machinery of deep results
of commutative algebra are applied to prove the final main result of
this note, Theorem 2.7. All of the rings which are considered in this
0

2010 Mathematics Subject Classification: 13A99, 13H99, 13B30.
Key words and phrases: flat topology, minimal prime.
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ABOLFAZL TARIZADEH

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

Uα where for each α, Uα is a flat open
S
subset of Spec(R). We claim that Spec(R) = Uα . Let p be a prime
α

α

ideal of R. There is a minimal prime p′ of R such that p′ ⊆ p. There
exists some α such that p′ ∈ Uα . We have p ∈ Uα . If not, then {p} ⊆
Uαc = Spec(R)\Uα . 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. 
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
[1, Tag 00DS],
S theorem
′
there is an element f ∈ p such that f ∈
/
p.
p′ ∈Min(R),
p′ 6=p

(ii) ⇒ (iii) : There is nothing to prove.
(iii) ⇒ (i) : It is an immediate consequence of Theorem 2.1. 
Theorem 2.1 vastly generalizes [7, Theorem 88], [5] and [6, Theorem
1.6]:
Corollary 2.3. Let R be a ring such that every minimal prime is the
radical 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.

ON THE FINITENESS OF MINIMAL AND MAXIMAL SPECTRA

3

Second proof. Here, we present another proof for it without using
the previous results. Let p be a minimal prime of R. By the hypothesis, 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.
S
We have Spec(R) =
V(p). The patch topology is compact, see
p∈Min(R)

[9, Proposition 2.4]. Therefore Min(R) is a finite set. 
The converse of Corollary 2.3 does not necessarily hold, see [4, Example 2.14].
Remark 2.4. Let {Ri : i ∈ I} be aQfamily of domains. For each i,
πi−1 (0) is a minimal prime of R =
Ri where πi : R → Ri is the
i∈I

canonical projection. Because, suppose there exists a prime ideal p of
R such that p ⊂ πi−1 (0). Pick a ∈ πi−1 (0) \ p. Then ab = 0 where
b = (δi,j )j∈I . It follows that b ∈ p, a contradiction. The minimal prime
πi−1 (0) is principal since it is generated by the sequence (1 − δi,j )j∈I .
Using this and Corollary 2.3Qthen we find a minimal prime p in every
infinite direct product ring Z such that p 6= πi−1 (0) for all i. Moreover, p is not the radical of aQf.g. ideal. It is worthy to mention that
the set of minimal primes of Z is in bijective correspondence to the
set 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 topology
over Spec(R). Then Max(R) the maximal spectrum of R is quasicompact.
Proof. It is well-known. Indeed, it is proved exactly like Theorem
2.1. 
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.
T
(ii) For each maximal ideal m, m +
m′ = R.
m′ ∈Max(R),
m′ 6=m

(iii) For each maximal ideal m there is an element f ∈ R \ m such that
{m} = Max(R) ∩ D(f ).

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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 Proposition
2.5. 
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 maximal
ideal m the canonical map π : R → Rm 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. Every
flat ring homomorphism which is also of finite presentation then it
induces a Zariski open map between the corresponding prime spectra,
see [1, Tag 00I1]. Therefore there are finite subsets {f1 , ..., fn } ⊆ R and
{m1 , ..., mn } ⊆ Max(R) such that R = (f1 , ..., fn ) and D(fi ) = Im πi∗
for all i where πi : R → Rmi is the canonical map. It follows that
Max(R) = {m1 , ..., mn }. 
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 Algebra, 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 a
commutative 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 [].
Department of Mathematics, Faculty of Basic Sciences, University
of Maragheh, P. O. Box 55136-553, Maragheh, Iran.
E-mail address: ebulfez1978@gmail.com

