Front. Math. ChinaDOI 10.1007/s11464-014-0388-0

Nagata rings

Pascual JARA

Department of Algebra, University of Granada, 18071–Granada, Spain

c© Higher Education Press and Springer-Verlag Berlin Heidelberg 2014

Abstract Let A be a commutative ring. For any set P of prime ideals ofA, we define a new ring Na(A,P) : the Nagata ring. This new ring has theparticularity that we may transform certain properties relative to P toproperties on the whole ring Na(A,P); some of these properties are: ascendingchain condition, Krull dimension, Cohen–Macaulay, Gorenstein. Our main aimis to show that most of the above properties relative to a set of prime idealsP (i.e., local properties) determine and are determined by the same propertieson the Nagata ring (i.e., global properties). In order to look for newapplications, we show that this construction is functorial, and exhibits afunctorial embedding from the localized category (A,P)-Mod into themodule category Na(A,P)-Mod.

Keywords Cohen–Macaulay, Gorenstein, Krull, Nagata ringsMSC 13BA15, 13B30, 13F20, 13F05, 16S85

1 Introduction

Let A be a commutative ring, and let P be a set of prime ideals of A. Wedeal with the properties of A induced by properties of the localized rings Ap,where p carries on P, as it was performed, for instance, in [5], [3], or [2],where, among others, Cohen–Macaulay and Gorenstein properties relative togenerically stable sets of prime ideals were studied.

It is well known that many classes of commutative rings may be describedin terms of the localization at certain sets of prime ideals. Let us exhibit someparticular examples.

(i) A commutative ring A is noetherian if(1) Ap is a noetherian local ring for any maximal ideal p and(2) for any finitely generated ideal a, the set of weak associated ideals

Assf (A/a) is finite.

Received September 17, 2013; accepted April 12, 2014E-mail: pjara@ugr.es

2 Pascual JARA

See [1,12].(ii) A commutative domain D is a Krull domain if(1) {Dp | p ∈ Spec(D) and ht(p) = 1} is a set of discrete valuation domains

and(2) D = ∩{Dp | p ∈ Spec(D) and ht(p) = 1} has finite character.

In particular, a Krull domain is a domain which is noetherian with respect tothe set Spec(D)1 = {p ∈ Spec(D) | and ht(p) = 1}.

These properties, which are defined in terms of the localized rings of theoriginal ring A, may be induced generally by a unifying concept: the hereditarytorsion theories, or equivalently, the localizing systems.

In the literature, there are several different methods for studying localproperties on commutative rings; one of the most effective is the study ofhereditary torsion theories. Let us show an example: let D be a Krull domain,there exists a hereditary torsion theory κ on D such that certain propertiesof the Krull domain D are parameterized by some properties of the localizedcategory (D,κ)-Mod, see Stenstrom’s book [16]. The category (D,κ)-Mod is areflexive full subcategory ofD-Mod and, in particular, a Grothendieck category;moreover, in general, the former category is not a module category.

Our aim in this work is to provide a new method for studying propertiesof a ring A, relative to a set of prime ideals of A, in terms of modules over anew ring. This method consists in the use of a ring, similar to the Nagata ring,as it was introduced in [15], but using a set P of prime ideals to define a newmultiplicative subset in the polynomial ring. Several constructions relativeto the original Nagata ring has been studied in particular contexts by otherauthors. The first example is the original one introduced by Nagata; heconsidered the case of a domain D and P = Spec(D), and used the polynomialring with coefficients in D either in one or several indeterminates. The secondone is the construction introduced by Fontana and Loper [9], among others (seealso [7] and [8]), in the framework of some generalization of Prufer domains,the Prufer �-Multiplication domains, with respect to a semistar operation.

In order to extend these constructions, we work on a commutative ring,not necessarily a domain, with a set P of prime ideals of A, and define thecorresponding multiplicatively closed subset in the polynomial ring in oneindeterminate, A[X], with coefficients in A. Our main result is to show thatthis new, and more general, Nagata ring parameterizes certain properties (asnoetherianness, Krull dimension, etc.) of the ring A with respect to the set Pof prime ideals. In fact, our aim in writing this paper is to show that undercertain finiteness conditions, the Nagata ring is a useful tool in the study oflocal properties of commutative rings.

The present work is divided into five sections. In Section 1, we introducethe background on localization at a set of prime ideals and a hereditary torsiontheory σ, and build the new Nagata ring. In Section 2, we show that the Nagataring of A and the Nagata ring of the localization Aσ of A are isomorphic. InSection 3, we define and study Nagata modules and show that this Nagata

Nagata rings 3

construction kills torsion modules performing the constructions in the contextof the localized category (A,σ)-Mod. In fact, a functor Na may be defined fromthe category (A,σ)-Mod to the category Na(A,σ)-Mod. Section 4 is devotedto the ascending chain condition. In it we show that the ring A is noetherianwith respect to the set of prime ideals, i.e., σ-noetherian, if and only if theNagata ring Na(A,σ) is noetherian. In Sections 5 and 6, we study several local-global properties of commutative rings and relates them to the Nagata ring. Wechoose several topics as Krull dimension, Krull domains, and Cohen–Macaulayor Gorenstein properties to show the behavior of the Nagata ring in the studyof the arithmetical and geometrical properties of commutative rings.

As background and main references, we use the classical books of Gilmer[10] and Kaplansky [14] for commutative rings; the books of Stenstrom [16] andGolan [11] for hereditary torsion theories, and the book of Fontana et al. [6],where hereditary torsion theories are also studied as localizing systems, forPrufer domains.

2 Nagata ring

Let A be a commutative (unitary) ring, and let σ be a hereditary torsion theoryin the category A-Mod of A-modules. Let us denote by Tσ (resp. Fσ) the classof σ-torsion (resp. σ-torsion-free) A-modules. To avoid trivial cases, throughoutthis paper, we assume σ �= 1, i.e., Tσ �= A-Mod and Fσ �= {0}.

Let M be an A-module, and let N ⊆ M a submodule of M. We say thatN is σ-dense in M if the quotient M/N is σ-torsion, and σ-closed in M if thequotient M/N is σ-torsion-free. If N ⊆M is σ-dense, we write N ⊆σ M.

Let N ⊆ M be a submodule. There exists a minimum σ-closed submoduleN of M containing N, we call it the σ-closure of N in M and sometimes werepresent it by ClMσ (N); in fact, N satisfies the identity N/N = σ(M/N).

We represent by L (σ) the set of all σ-dense ideals of A. It is well knownthat any hereditary torsion theory is defined by the set L (σ) as we have

Tσ = {M ∈ A-Mod | AnnA(m) ∈ L (σ) for any m ∈M}.We represent by K (σ) the set of all σ-closed prime ideals and by C (σ) the setof all maximal elements in K (σ). For any A-module M, we represent by Cσ(M)the lattice of all σ-closed submodules of M, where the meet of L1, L2 ∈ Cσ(M)is defined as the intersection and the join as L1 + L2, the σ-closure of L1 +L2.

Let σ be a hereditary torsion theory, let us consider the following definitions:(1) σ is half-centered, if for each ideal a ⊆ A such that a /∈ L (σ), there

exists a σ-closed prime ideal p ⊆ A such that a ⊆ p, see [4], and(2) σ is of finite type, if L (σ) has a cofinite subset of finitely generated

ideals. It is well known that each finite type hereditary torsion theory is half-centered.

Let p ∈ Spec(A) be a prime ideal. There exists a hereditary torsion theory

4 Pascual JARA

σA\p, defined by p as follows:

L (σA\p) = {a ⊆ A | a � p}.As a consequence, we have

(1) σA\p is a finite type, hence, a half-centered hereditary torsion theory,and

(2) a hereditary torsion theory σ is half-centered if and only if it satisfies

σ = ∧{σA\p | p ∈ K (σ)},where the meet ∧σA\p is defined as the hereditary torsion theory such that

T∧σA\p= ∩TσA\p

.

As we point out before, any finite type hereditary torsion theory σ is half-centered. In addition, a half-centered hereditary torsion theory is of finite typeif and only if the set of prime ideals K (σ) is quasi-compact in Spec(A) wheneverwe consider on Spec(A) the Zariski topology.

If we start from a (nonempty) set of prime ideals P, there exists a hereditarytorsion theory σ defined by P as follows:

σ = ∧{σA\p | p ∈ P}.In that case, we have

P ⊆ K (σ) = P↓ := {q ∈ Spec(A) | there is p ∈ P such that q ⊆ p}.For any polynomial f ∈ A[X], we define the content of f , as the ideal of

A generated by the coefficients of f, and represent it by c(f). Let a ⊆ A bean ideal. We represent by a[X] the ideal of A[X] generated by all polynomialswith coefficients in a.

Lemma 2.1 Let σ be a half-centered hereditary torsion theory in the categoryA-Mod. Then the subset

Σ(σ) = {f ∈ A[X] | c(f) ∈ L (σ)} ⊆ A[X]

is multiplicatively closed in A[X].

Proof We show that Σ(σ) is the complement of the union ∪{p[X] | p ∈ K (σ)}.Indeed,

f ∈ Σ(σ) ⇐⇒ c(f) = A

⇐⇒ c(f) � p for all p ∈ K (σ)⇐⇒ f ∈ A[X] \ p[X] for all p ∈ K (σ)

⇐⇒ f ∈⋂

p∈K (σ)

(A[X] \ p[X]) = A[X] \⋃

p∈K (σ)

p[X].

Nagata rings 5

Now, Σ(σ) is the complement of a union of prime ideals, and hence, it ismultiplicatively closed in A[X]. �

Let us represent the ring A[X]Σ(σ) simply by Na(A,σ) and call it the Nagataring of A relative to σ. In the particular case, in which σ is the hereditarytorsion theory defined by a set P of prime ideals, we may also write Na(A,P).

In order to study Na(A,σ), we shall show that the set of its maximal idealsis exactly

{p[X]Σ(σ) | p ∈ C (σ)}.Lemma 2.2 Let a ⊆ A[X] be an ideal, let h ⊆ A be the ideal generated by allcoefficients of polynomial in a, and let h′ ⊆ h be a finitely generated ideal. Thenthere exists f ∈ a such that h′ ⊆ c(f).

Proof We claim that for any a ∈ h, there exists f ∈ a such that a is a coefficientof f. Indeed, if a =

∑ti=1 riai, where ri ∈ A and ai is a coefficient of fi ∈ a,

then riai is a coefficient of rifi, and we may assume that each ri is equal to 1.When ai is the coefficient of degree mi of fi, we define

m = max{m1, . . . ,mt}

and define

f =t∑i=1

Xm−mifi.

Then a is the coefficient of degree m of f. Let h′ = (a1, . . . , as) ⊆ h, and foreach i = 1, . . . , s, let fi ∈ a such that ai is a coefficient of fi; if degree(fi) = ni,we define a new polynomial f ∈ a as follows:

f = f1 +Xn1+1f2 + · · · +Xn1+···+ns−1+1fs.

In this way, h′ ⊆ c(f).

Proposition 2.3 Let σ be a finite type hereditary torsion theory in A-Mod.Then {p[X]Σ(σ) | p ∈ C (σ)} is the set of all maximal ideals of Na(A,σ), i.e.,each proper ideal q ⊆ Na(A,σ) is contained in one of the p[X]Σ(σ).

Proof We know that {p[X] | p ∈ K (σ)} is a set of prime ideals of A[X]contained in A[X]\Σ(σ). Then we shall show that each proper ideal of Na(A,σ)is contained in one of the p[X]Σ(σ). Let b ⊂ Na(A,σ) be a proper ideal. Thenb = aNa(A,σ) for some a ⊆ A[X] such that a ∩ Σ(σ) = ∅; then

a ⊆ ∪{p[X] | p ∈ K (σ)}.Let h ⊆ A be the ideal generated by the coefficients of polynomials in a. Sincea ⊆ ∪p[X], we have h /∈ L (σ). Indeed, if h ∈ L (σ), then there is h′ ⊆ h finitelygenerated and h′ ∈ L (σ). By Lemma 2.2, we may assume h′ = c(f) for somef ∈ a. Since a ⊆ ∪p[X], there is p such that f ∈ p[X], and therefore, p ∈ L (σ),which is a contradiction. Then h /∈ L (σ) and there is p ∈ K (σ) such that

6 Pascual JARA

h ⊆ p. Moreover, a ⊆ p[X]. As a consequence, the maximal ideals of Na(A,σ)have the form p[X]Σ(σ) for p ∈ K (σ). Therefore, p[X]Σ(σ) is a maximal idealof Na(A,σ) if and only if p ∈ K (σ) is maximal. �

In order to obtain the properties on the behavior of the Nagata ring, we needsome technical results on hereditary torsion theories relative to the change ofring.

Let ϕ : A → B be a ring homomorphism between commutative rings. Ifσ is a hereditary torsion theory in A-Mod, we define a new hereditary torsiontheory, say ϕ(σ), in B-Mod. In fact, if we define

L (ϕ(σ)) = {b ⊆ B | ϕ−1(b) ∈ L (σ)},Tϕ(σ) = {M ∈ B-Mod |A M ∈ Tσ},Fϕ(σ) = {M ∈ B-Mod |A M ∈ Fσ},

then we have the following result.

Proposition 2.4 With the above notation, ϕ(σ) is a hereditary torsion theoryin B-Mod with torsion class Tϕ(σ), torsion-free class Fϕ(σ), and Gabriel filterL (ϕ(σ)). �Lemma 2.5 Let σ be a finite type hereditary torsion theory. Then ϕ(σ) is offinite type.

Proof Let b ∈ L (ϕ(σ)). Since ϕ−1(b) ∈ L (σ), there exists a finitely generatedideal a ∈ L (σ) such that a ⊆ ϕ−1(b), and then, ϕ(a)B ⊆ b and ϕ(a)B ∈L (ϕ(σ)) is finitely generated. �

Let σ be a hereditary torsion theory in A-Mod. For any A-module M,we define a new A-module, say Mσ, called the localization of M at σ; see[16, Chapter XX], in such a way that Aσ is a commutative ring and Mσ isan Aσ-module. For any A-module M, there exists a map ϕM : M → Mσ withkernel σ(M), the maximal submodule of M contains in Tσ.

Proposition 2.6 Let ϕ := ϕA : A→ Aσ be the canonical map. Then for anyA-module M,

(1) there is a natural isomorphism (Mσ)ϕ(σ)∼= Mσ;

(2) there is a lattice isomorphism Cσ(M) → Cϕ(σ)(Mσ) given by

Cσ(M)β−→ Cϕ(σ)(Mσ),

X �→ X + σ(M)σ(M)

�→ β(X) := Xσ,

with inverse map γ : Y �→ ϕ−1(Y ).(3) In the above bijection, if we consider M = A, then ideals in C (σ)

correspond with ideals in C (ϕ(σ)).

Nagata rings 7

Proof We shall prove only part (3). In fact, elements in C (σ) are the maximalelements in Cσ(A). �

Let us apply now these results to the localization at prime ideals p ∈ K (σ).

Proposition 2.7 For any hereditary torsion theory σ in A-Mod and anyprime ideal p ∈ K (σ), the map ϕ : A→ Aσ induces the identity

ϕ(σA\p) = σAσ\pσ.

Proof We show that both hereditary torsion theories have the same torsionmodules. Let Z be a σAσ\pσ

-torsion Aσ-module. Then for any z ∈ Z, thereis s ∈ Aσ \ pσ such that sz = 0. Since s ∈ Aσ , there exists a ∈ L (σ) suchthat as ⊆ A. If as ⊆ p, then s ∈ pσ, which is a contradiction. Therefore, thereexists y ∈ a such that ys /∈ p and satisfies ysz = 0. Hence, Z is ϕ(σA\p)-torsion.Otherwise, let Z be a ϕ(σA\p)-torsion Aσ-module. Then for any z ∈ Z, thereexists s ∈ A \ p such that sz = 0. Since s ∈ Aσ \ pσ, Z is σAσ\pσ

-torsion. �Lemma 2.8 Let p ∈ K (σ) be a prime ideal. Then, for any A-module M, wehave

(Mσ)p ∼= Mp∼= (Mσ)pσ .

Proof We assume that M is σ-torsion-free. Since σ � σA\p, we have acommutative diagram

σA\p �� ����

��

σA\p(Mσ) ����

��

σA\p(Mσ/M)

M�� β ��

��

�������

����

� M� ��

��

��������

������

Mσ/M

��

�������������

• �� ��

��

• ����

����������� 0

Mp�� βp �� (Mσ)p �� 0

Then β has σA\p-torsion cokernel, and hence, βp is surjective. Therefore, it isan isomorphism and we have Mp

∼= (Mσ)p.We know that if p ∈ C (σ), then pσ ∈ C (ϕ(σ)). On the other hand, using

Proposition 2.7, we have

(Mσ)pσ∼= (Mσ)σAσ\pσ

∼= (Mσ)ϕ(σA\p)∼= (Mσ)σA\p

∼= Mp. �

The last result in this section deals with σ-dense ideals.

Lemma 2.9 Let a ⊆ b be ideals of A. Then the following statements hold.(1) If a ⊆σ b, then aAσ ⊆ϕ(σ) bAσ.

(2) Let σ be a hereditary torsion theory of finite type. If aAσ ⊆ϕ(σ) bAσ,then a ⊆σ b.

8 Pascual JARA

(3) If aAσ = bAσ, then a ⊆σ b.

Proof (1) For any y ∈ b, there is c ∈ L (σ) such that cy ⊆ a. Since c ⊆ϕ−1(cAσ), we have

cAσ ∈ L (ϕ(σ)), cAσy ⊆ aAσ,

and then,y ∈ aAσ, aAσ ⊆ϕ(σ) aAσ .

(2) For any y ∈ b, there is c ∈ L (σ), finitely generated, such that cy ⊆ aAσ .Then there is c′ ∈ L (σ) such that c′cy ⊆ a. As a consequence, a ⊆σ b.

(3) For any y ∈ b, there exist ai ∈ Aσ and xi ∈ a, i = 1, . . . , t, such thaty =

∑i aixi. Then there is c ∈ L (σ) such that cai ∈ A for any index i, and

then cy ∈ a and a ⊆σ b. �

3 Invariance of Nagata ring

We have established before the definition of the Nagata ring with respect toa half-centered hereditary torsion theory. From the definition, we deduce thatdifferent rings may define the same Nagata ring, and now we delve into thissituation; in fact, we shall show that A and Aσ define the same Nagata ring.

We have the following commutative diagram:

Aπ ��

ϕ

��

A[X] λ ��

ψ��

Na(A,σ)

φ�����

Aσπσ �� Aσ[X]

λσ �� Na(Aσ , ϕ(σ))

(3.1)

where π and πσ are the canonical inclusions, and λ and λσ the canonical mapsfrom a ring in it localized at a multiplicatively closed subset. We call ψ themap between the polynomial rings induced by ϕ. By the definition, we have thefollowing identities:

Σ(ϕ(σ)) = {f ∈ Aσ[X] | c(f) ∈ L (ϕ(σ))}= {f ∈ Aσ[X] | ϕ−1(c(f)) ∈ L (σ)}.

If f ∈ Σ(σ), then c(f) ∈ L (σ), and hence,

c(ψ(f)) = ϕ(c(f))Aσ ⊆ϕ(σ) Aσ.

Therefore, ψ(f) ∈ Σ(ϕ(σ)). Moreover, there is a unique ring homomorphism

φ : Na(A,σ) → Na(Aσ, ϕ(σ))

such that diagram (3.1) commutes.

Nagata rings 9

Lemma 3.1 Let σ be a half-centered hereditary torsion theory in the categoryA-Mod, and let ϕ : A → Aσ and ψ : A[X] → Aσ[X] be the canonical maps.Then for any polynomial f ∈ Σ(σ), we have ψ(f) �= 0.

Proof Let f ∈ Σ(σ). If ψ(f) = 0, then c(f) ⊆ σ(A), hence, σ(A) ∈ L (σ),which implies σ(A) = A, and therefore, σ = 1, which is a contradiction. �Theorem 3.2 Let σ be a half-centered hereditary torsion theory in A-Mod.Then there exists a ring isomorphism φ : Na(A,σ) → Na(Aσ , ϕ(σ)).

Proof Both of them Na(A,σ) and Na(Aσ, ϕ(σ)) are Na(A,σ)-modules. Sincewhen we localize at each maximal ideal q of Na(A,σ), or equivalently, at eachprime ideal p ∈ C (σ), obtain identities, we get that φ is an isomorphism. Letus assume

q = p[X]Σ(σ).

ThenΣ(σ) ⊆ A[X] \ p[X].

Indeed, if f ∈ Σ(σ), then

c(f) ∈ L (σ) ⊆ L (σA\p), c(f) � p, f /∈ p[X].

As a consequence,

(A[X]Σ(σ))q = (A[X]Σ(σ))p[X]Σ(σ)= A[X]p[X].

Since ψ(Σ(σ)) ⊆ Σ(ϕ(σ)), there exists a map

φ : A[X]Σ(σ) → Aσ[X]Σ(ϕ(σ)) .

The maximal ideals of Aσ[X]Σ(ϕ(σ)) are pσ[X]Σ(ϕ(σ)) , where p runs in C (σ).Then for any q = p[X]Σ(σ), we have

(Aσ[X]Σ(ϕ(σ)))q = (Aσ [X]Σ(ϕ(σ)))p[X]Σ(σ)= (Aσ[X]Σ(ϕ(σ)))p[X]

as ψ(Σ(σ)) ⊆ Σ(ϕ(σ)) implies

(Aσ [X]Σ(ϕ(σ)))Σ(σ) = Aσ[X]Σ(ϕ(σ)).

Since localize at p[X] is the same that (localize) at ψ(σA[X]\p[X]), todetermine ψ(σA[X]\p[X]), first we observe that pσ [X] ∈ K (ψ(σA[X]\p[X])), andsecond we consider the definition of L (ψ(σA[X]\p[X])), i.e., b ∈ L (ψ(σA[X]\p[X]))if and only if ψ−1(b) ∈ L (σA[X]\p[X]), if and only if ψ−1(b) � p[X]. Therefore,for any h /∈ L (ψ(σA[X]\p[X])) and any f ∈ h, there exists c ∈ L (σ) such that

cc(f) ⊆ ϕ(A), cf ⊆ h ∩ ψ(A[X]) ⊆ φ(p[X]).

Then f ∈ pσ [X]. Thus, pσ[X] is the only maximal element in K (ψ(σA[X]\p[X])).Always, we obtain

ψ(σA[X]\p[X]) = σAσ[X]\pσ[X].

10 Pascual JARA

Indeed, if Z is σAσ [X]\pσ[X]-torsion, then for any z ∈ Z, we have

Ann(z) � pσ [X].

Therefore, there exists

f ∈ AnnAσ [X](z) \ pσ[X].

Since f /∈ pσ[X], we have c(f) � pσ and there exists c ∈ L (σ) such that

cc(f) ⊆ A \ p.

Then cf ⊆ A[X] \ p[X] and cf ⊆ AnnAσ[X](z). Hence, AnnAσ [X](z) � p[X] andz ∈ σA[X]\p[X]. In this case, we have

(Aσ[X]Σ(ϕ(σ)))p[X] = (Aσ [X]Σ(ϕ(σ)))pσ [X]

= (Aσ [X]Σ(ϕ(σ)))pσ [X]Σ(ϕ(σ))

= Aσ[X]pσ [X]

= Aσ[X]p[X].

Now, the natural map

φp[X] : A[X]p[X] → Aσ[X]p[X]

is an isomorphism as if f ∈ Aσ[X], there is c ∈ L (σ) such that cc(f) ⊆ A, andthen cf ⊆ ψ(A[X]). Since c ∈ L (σ), we have c � p, and there is y ∈ c\p. Hence,ϕ(y) �= 0, as if ϕ(y) = 0, then y ∈ σ(A) ⊆ σA\p(A), which is a contradiction.In this case, we can consider g = yf ∈ ψ(A[X]) and get

f = g/y ∈ φp[X](A[X]p[X]).

Hence, φp[X] is surjective. �We obtain that Na(A,σ) is a new ring in which the σ-dense ideals of A

disappear and such that the maximal ideals are in bijective correspondencewith the elements in C (σ), i.e., the set of the maximal σ-closed ideals of A.

Lemma 3.3 Letν : A π−→ A[X] λ−→ Na(A,σ)

be the composition of the canonical ring maps. Then the hereditary torsiontheory ν(σ), induced by σ in Na(A,σ) via ν, is the trivial one.

Proof Let b ⊂ Na(A,σ) such that ν−1(b) ∈ L (σ). There exists a maximalideal q ⊆ Na(A,σ) such that b ⊆ q. Then

ν−1(b) ⊆ ν−1(q) ∈ C (σ),

which is a contradiction. �

Nagata rings 11

4 Nagata modules

In this section, we shall study the behavior of finite type hereditary torsiontheories with respect to the Nagata construction. First, we extend the Nagataconstructions from rings to modules.

Let M be an A-module. We define the Nagata module of M with respect toσ as

Na(M,σ) = M [X]Σ(σ) = (M ⊗A A[X])Σ(σ).

This construction allows us to define a functor

Na(−, σ) : A-Mod → A-Mod,

and we may also consider this functor with image in the module categoryNa(A,σ)-Mod.

Lemma 4.1 Let M be an A-module. If Na(M,σ) = 0, then M is σ-torsion.

Proof Let m ∈ M. Then there exists g ∈ Σ(σ) such that gm = 0. Hence,c(g) ⊆ Ann(m). Since g ∈ Σ(σ), Ann(m) ∈ L (σ). Therefore, m ∈ σ(M). �Lemma 4.2 Let σ be a finite type torsion theory, and let M be a σ-torsionA-module. Then Na(M,σ) = 0.

Proof Let f ∈ M [X]. Then there is c ∈ L (σ) such that cc(f) = 0. We cantake c finitely generated. Then there is g ∈ A[X] such that c(g) = c, and then,g ∈ Σ(σ) and gf = 0. As a consequence, f/1 = 0 in Na(M,σ). �Corollary 4.3 Let σ be a finite type torsion theory, and let ω : M → N be anA-map. Then the following statements are equivalent :

(a) ω has torsion kernel and cokernel,(b) Na(ω, σ) : Na(M,σ) → Na(N,σ) is an isomorphism.

Proof We only need to use that Na(−, σ) is a functor which is the compositionof the exact functors −⊗A A[X] and (−)Σ(σ). �

Our aim is to extend the isomorphism φ : Na(A,σ) → Na(Aσ , ϕ(σ)) to anisomorphism between the Nagata modules of an A-module M. If M is an A-module, we have a map γ : M ⊗A Aσ → Aσ defined by γ(m ⊗ x) = ϕM (m)x.Hence, we may consider the following diagram:

Na(M,σ)1⊗Aϕ ��

α

��

Na(M ⊗A Aσ, ϕ(σ))γ ��

�

Na(Mσ, ϕ(σ))

�

M ⊗A Na(A,σ)1⊗Aφ �� M ⊗A Na(Aσ , ϕ(σ))

ϕM⊗1�� Mσ ⊗Aσ Na(Aσ , ϕ(σ))

where “overline” means to apply the functor either Na(−, σ) or Na(−, ϕ(σ)).Since φ is an isomorphism, 1 ⊗ φ is also an isomorphism, and since the kerneland cokernel of γ are ϕ(σ)-torsion, γ is an isomorphism. Otherwise, by the

12 Pascual JARA

definition of the Nagata module, α, β, and γ are isomorphisms and the diagramcommutes. Thus, we obtain the following theorem.

Theorem 4.4 Let σ be a finite type hereditary torsion theory in A, and letM be an A-module. Then there exists a natural isomorphism

γ ◦ 1 ⊗A ϕNa(M,σ) ∼= Na(Mσ , ϕ(σ))

of Na(A;σ)-modules.

Once we know how modules work with respect to the Nagata constructions,we may turn to consider the behavior of the ideals of the ring A, as it is similarto the behavior of modules. Easily, we have the following result.

Lemma 4.5 Let σ be a finite type hereditary torsion theory in A such thatσ(A) = 0. For any ideal a ⊆ A, we have

aNa(A,σ) ∩A = ClAσ (a) =: a.

Proof Since a is σ-dense in a, we have

aNa(A,σ) = Na(a, σ) = Na(a, σ) = a Na(A,σ),

and thena ⊆ a ⊆ aNa(A,σ).

Otherwise, if f ∈ aNa(A,σ) ∩A[X], then there exists h ∈ L (σ) such thathf ∈ a. Hence, c(hf) ⊆ a. Then

a ⊇ c(hf) = c(h)mc(hf) = c(h)m+1c(f) = c(f) ⊇ c(f),

and we obtain f ∈ aA[X]. Now, since aA[X] ∩A = a, the result holds. �Example 4.6 Let A be a ring and consider 0 the trivial hereditary torsiontheory, i.e., every A-module is torsionfree. Then Na(A, 0) is the Nagata ring asit was introduced in [15]. We represent it simply by A(X). In the same way,we represent Na(M, 0) by M(X) for any A-module M.

Remark 4.7 Let A = K be a field. Then the Nagata ring Na(K, 0) = K(X)is exactly the field of fractions of the polynomial ring K[X]. Let M be aK-vector space. Then the Nagata module M(X) is a K(X)-vector space. Wehave M(X) is always infinite dimension as K-vector space. It is easy to showthat the functor Na(−, σ) has no fixed points, i.e., in general, there are notnontrivial A-modules M such that Na(M,σ) = M.

5 Noetherian case

Even if Na(M,σ) is big enough as an A-module, the Na(A,σ)-module structureof Na(M,σ) can be easily controlled in certain particular cases.

Nagata rings 13

Let A be a ring. As in Example 4.6, we consider the trivial hereditarytorsion theory 0 on A, i.e., 0(M) = 0 for every A-module M. We representNa(A, 0) simply by either A(X) or Na(A), the Nagata ring of A, and call Σ(0)simply Σ.

Lemma 5.1 Let A be a (local) noetherian ring. Then A(X) is a (local)noetherian ring.

Here, local means that the ring A has only one maximal ideal.

Proof of Lemma 5.1 Since A is noetherian, A[X] is noetherian, and A(X) isthe localization at a multiplicatively closed subset is also noetherian.

If A is in addition local with maximal ideal m, then C (0) = {m}, and hence,A(X) is a local ring. �Lemma 5.2 Let A be a local ring. If A(X) is noetherian, then A isnoetherian.

Proof Leta1 ⊆ a2 ⊆ a3 ⊆ · · ·

be a chain of ideals of A. Then

a1A(X) ⊆ a2A(X) ⊆ a3A(X) ⊆ · · ·

is a chain of ideals of A(X), and there is an index n such that

anA(X) = an+1A(X) = · · · .

For any x ∈ an+1, there are yi ∈ an and fi/g ∈ A(X) such that

x =t∑i=1

yifi/g.

There is h ∈ Σ such that

xgh =t∑i=1

yifih.

Since gh ∈ Σ, we have c(gh) = A, and hence,

x ∈ xA = xc(gh) = c(xgh) ⊆ c( t∑i=1

yifih

)⊆ an.

Therefore, x ∈ an. �From the above results, we have the Nagata construction passes local

information up (from A to Na(A)) and down (from Na(A) to A). But theseresults can be also extended to the local-global case as follows.

Let us first show a technical lemma.

14 Pascual JARA

Lemma 5.3 Let a ⊆ A be an ideal and A = Aσ. Then the following statementshold:

(1) x a ⊆ xa for any x ∈ A;(2) ac = a for any c ∈ L (σ).

Proof (1) If for z ∈ a there exists c ∈ L (σ) such that cz ⊆ a, then cxz ⊆ xaand we obtain xz ∈ xa.

(2) Since ac ⊆ a, we obtain ac ⊆ a. Otherwise, if x ∈ a, then there existsc′ ∈ L (σ) such that c′x ⊆ a. We have cc′x ⊆ ac, and hence, x ∈ ac. �Theorem 5.4 Let σ be a hereditary torsion theory in A. Then the followingstatements are equivalent :

(a) A is σ-noetherian;(b) Na(A,σ) is noetherian.

Proof (a) =⇒ (b). Let K ⊆ Na(A,σ) be an ideal. Then there is b ⊆ A[X] suchthat bΣ(σ) = K, and there is b′ ⊆ b finitely generated such that b′ ⊆ π(σ)b asA is σ-noetherian, then A[X] is π(σ)-noetherian. Then K = (b′)Σ(σ) is finitelygenerated.

(b) =⇒ (a). In this case, we may assume A = Aσ. Let a ⊆ A be an ideal.Since Na(a, σ) ⊆ Na(A,σ) is finitely generated, there exists b ⊆ A[X] finitelygenerated such that Na(a, σ) = bΣ(σ). Then for any g ∈ a[X], there are g′ ∈ band g′′ ∈ Σ(σ) such that gg′′ = g′g′′. In particular, for any x ∈ a, there arex′ ∈ b and x′′ ∈ Σ(σ) such that xx′′ = x′x′′. Then

x ∈ xA = x c(x′′) ⊆ xc(x′′) = c(xx′′) = c(x′x′′) ⊆ c(x′)c(x′′) = c(x′)

If we call h the ideal of A generated by the coefficients of all polynomials in b,then x ∈ h. Otherwise, since Na(a, σ) = bΣ(σ), for any h ∈ b, there are h′ ∈ a[X]and h′′ ∈ Σ(σ) such that hh′′ = h′h′′. Let z ∈ h. Then there exists h ∈ b suchthat z is a coefficient of h. Hence, there are h′ ∈ a[X] and h′′ ∈ Σ(σ) such thathh′′ = h′h′′. Using [10, Theorem 28.1], there exists a positive integer m suchthat

c(h′′)m+1c(h) = c(h′′)mc(hh′′) = c(h′′)mc(h′h′′) = c(h′′)m+1c(h′).

Then c(h) = c(h′) and we obtain

z ∈ c(h) ⊆ c(h) ⊆ c(h′) ⊆ a.

Therefore, we obtain the inclusions

a ⊆ h ⊆ a.

As a consequence, a contains a finitely generated σ-dense ideal. �

Nagata rings 15

6 Examples and applications

This final section is devoted to showing that some properties are preserved bythe Nagata construction.

6.1 Krull domains

Let D be a Krull domain, and let κ the hereditary torsion theory defined by

Spec1(D) := {p ∈ Spec(D) | ht(p) = 1}.We have D is a κ-noetherian domain. Hence, the Nagata ring Na(D,κ) of Drelative to κ is a noetherian domain. In fact, Na(D,κ) is a Dedekind domain.

Proposition 6.1 Let D be a domain, and let κ be the hereditary torsion theorydefined by Spec1(D). Then the following statements are equivalent :

(a) D is a Krull domain;(b) Na(D,κ) is a Dedekind domain.

Proof By definition, κ is half-centered. Hence, we may define the Nagata ringNa(D,κ) andD is κ-noetherian if and only if Na(D,κ) is noetherian. Otherwise,let p ∈ Spec1(D) be a height one prime ideal. Then Na(p, κ) is a prime ideal ofNa(D,κ) and

Na(D,κ)Na(p,κ)∼= Dp(X)

is a discrete valuation domain if and only if Dp is a discrete valuation domain,see [10]. �

In particular, the class group of D and the class group of Na(D,κ) areisomorphic.

6.2 Dimension

Let σ be a hereditary torsion theory in A such that A is σ-noetherian ring,i.e., the lattice Cσ(A) satisfies the ascending chain condition. We define theσ-dimension of A as the supremum of the lengths of chains of prime idealsin K (σ), and if M is an A-module and π : A → A/Ann(M) is the canonicalprojection, we define the σ-dimension of M as the π(σ)-dimension of the ringA/Ann(M), i.e., the supremum of lengths of chains of prime ideals in the setV (Ann(M)) ∩ K (σ). We represent it by dimσ(M).

For any A-module M, the σ-support of M is

Suppσ(M) = {p ∈ K (σ) |Mp �= 0}.Let M be a σ-finitely generated A-module, i.e., there exists a finitely generatedσ-submodule N ⊆M. Then

Suppσ(M) = V (Ann(M)) ∩ K (σ).

In this particular case, the σ-dimension of M is the dimension of the setSuppσ(M), see [3].

16 Pascual JARA

Lemma 6.2 Let M be a σ-finitely generated A-module, and let σ be ahereditary torsion theory in A such that A is a σ-noetherian ring. Then wehave

dimσ(M) = dim(Na(M,σ)).

Proof In fact, we have

dimσ(M) = sup{dimAp (Mp) | p ∈ Suppσ(M)}.Hence, the problem has a local nature. We may assume that A is a localnoetherian ring with maximal ideal p and M is a faithful A-module and needto show

dim(A(X)) = ht(p) = dim(A).

Indeed, the maximal ideal p(X) of A(X) is defined by p[X], which is notmaximal in A[X]. Let q ⊇ p[X] be a maximal ideal in A[X]. Since qΣ(0) = A(X),we have

dim(A(X)) = ht(p(X)) = ht(p) = dim(A). �Corollary 6.3 Let A be a σ-noetherian ring. Then

dimσ(A) = dim(Na(A,σ)).

Remark 6.4 Let A be a local ring with maximal ideal p, and let σ be thehereditary torsion theory σA\p. Let

p0 ⊂ · · · ⊂ pt = p

be a saturated chain of prime ideals in A, and let

h1 ⊂ · · · ⊂ hs ⊂ p[X] ⊆ h

be a chain of prime ideals in A[X]. Then either Na(p, σ) = hΣ(σ) or hΣ(σ) =Na(A,σ). Hence, the dimension of Na(A,σ) is bounded as follows:

dim(A) + 1 � dim(Na(A,σ)) � 2 dim(A).

Indeed, if q0 ⊂ q1 ⊂ p[X] is a chain of prime ideals in A[X], then

(q0)Σ(σ) ⊂ (q1)Σ(σ) ⊂ Na(p, σ).

6.3 Zero dimensional rings

Let A be a ring, and let P ⊆ Spec(A) be a nonempty set of minimal primeideals. Let us call σ the hereditary torsion theory defined by P. If σ is of finitetype, then we have the Nagata ring Na(A,σ) is 0-dimensional. We shall showthat the converse is also true. In fact, the converse passes by characterizingwhen a set of minimal prime ideals is quasi-compact; this is one of the mostimportant problems in ring with zero divisors as it is pointed out in Huckaba’sbook [13].

Nagata rings 17

In order to find this characterization, we define a ring to be reduced ringwith respect to P if 0 = ∩{p | p ∈ P}. If A is a reduced ring with respect toP and P is quasi-compact, then we have σ is a finite type hereditary torsiontheory, and hence, the set of maximal ideals of Na(A,σ) is {Na(p, σ) | p ∈ P},and Na(A,σ) is 0-dimensional. On the other hand, if A is reduced with respectto P and Na(A,σ) is 0-dimensional, then we have the following facts.

(1) Na(A,σ) is a reduced ring. Indeed,

∩{Na(p, σ) | p ∈ P} = Na(∩{p | p ∈ P}, σ) = 0.

(2) Na(A,σ) is a von Neumann regular ring and every finitely generatedideal is principal generated by an idempotent. See [13, p. 5].

(3) The set of maximal ideals in Na(A,σ) is {Na(A,σ) | p ∈ P}. Indeed, ifq ⊆ Na(A,σ) is a maximal ideal, then q = hΣ(σ) for some prime ideal h ⊆ A[X]such that h ∩ Σ(σ) = ∅. Let p = h ∩ A, then Na(p, σ) ⊆ q, and since Na(A,σ)is 0-dimensional, q = Na(p, σ).

(4) P is quasi-compact in Spec(A). Indeed, let P ⊆ ∪λX(aλ) be a unionof open sets in Spec(A). Then P ⊆ X(

∑λ aλ). Therefore,

∑λ aλ � p for any

p ∈ P, then

Na(∑

λ

aλ, σ

)� Na(p, σ),

and hence,

Na(∑

λ

aλ, σ

)= Na(A,σ).

Therefore, there exist indices λ1, . . . , λt, such that

Na(aλ1 + · · · + aλt , σ) = Na(A,σ),

and hence,aλ1 + · · · + aλt ∈ L (σ),

i.e.,P ⊆ X(aλ1 + · · · + aλt) = X(aλ1) ∪ · · · ∪X(aλt).

Thus, we have proved the following result.

Theorem 6.5 Let P be a set of minimal ideal of a commutative ring Asuch that A is reduced with respect to P. Then the following statements areequivalent :

(a) P is quasi-compact in Spec(A);(b) Na(A,σ) is 0-dimensional;(c) Na(A,σ) is a von Neumann regular ring.In this case, there exists a bijection between prime ideals in P, prime ideals

in the localized ring Aσ, and prime ideals in the Nagata ring Na(A,σ).

18 Pascual JARA

6.4 One dimensional rings

In point (3) in the proof of Theorem 6.5, we showed that for every prime idealq in Na(A,σ), there exists a prime ideal p in P such that q = Na(p, σ). Thisresult is also true in others situations, let us show another one which shall beof interest in further studies on Krull domains.

Theorem 6.6 Let P be a set of prime ideals of height less or equal than 1,let σ be the hereditary torsion theory defined by P, and let A be σ-noetherian.Then for every prime ideal q in Na(A,σ), there exists a prime ideal p ∈ P↓such that q = Na(p, σ).

Proof We may assume that A is a local noetherian domain with maximal idealm, and σ is the hereditary torsion theory defined by m. Let q ⊆ Na(A,σ) be aprime ideal. Then there exists a prime ideal h ⊆ A[X] such that

hΣ(σ) = q, h ∩ Σ(σ) = ∅.If we define p = h ∩A, then we have a chain of prime ideals

Na(p, σ) ⊆ hΣ(σ) = q ⊆ Na(m, σ).

Since dim(Na(A,σ)) = 1, we have either q = Na(p, σ) or q = Na(m, σ). �6.5 Cohen–Macaulay, Gorenstein, and regular rings

We give a few words to show the geometrical behavior of the Nagata ring. LetP be a set of prime ideals of a commutative ring A. We say that A is Cohen–Macaulay with respect to P if A is σ-noetherian (where σ is the hereditarytorsion theory defined by P) and Ap is a local Cohen–Macaulay ring for anyprime ideal p ∈ P. In the same way, we may define a ring to be either Gorensteinor regular with respect to P.

Theorem 6.7 Let A be a reduced commutative ring, and let P be a set ofprime ideals such that A is σ-noetherian. Then the following statements areequivalent :

(a) A is Cohen–Macaulay with respect to P.

(b) Na(A,σ) is Cohen–Macaulay.

Proof (a) =⇒ (b). This is a consequence of the facts that polynomials andlocalizations preserve the Cohen–Macaulay property.

(b) =⇒ (a). Since A is σ-noetherian, the maximal ideals of Na(A,σ) areNa(p, σ), where p varies over P. If we localize Na(A,σ) at Na(p, σ), then weobtain the ring A[X]p[X], and this is the Nagata ring of Ap at the maximalideal pAp. Hence, the result is a consequence of the following equivalence forany reduced local noetherian ring B :

B is Cohen–Macaulay ⇐⇒ B(X) is Cohen–Macaulay.

If B(X) is Cohen–Macaulay and m is the maximal ideal of B, then there exists amaximal B(X)-sequence f1, . . . , fn in m(X) with n = dim(B(X)). We assume

Nagata rings 19

fi = gi/1 where gi ∈ m[X] for i = 1, . . . , n. We claim that if f1 is non zerodivisor, then g1 is non zero divisor. Indeed, if hg1 = 0, then

(h/1)(g1/1) = 0,

and hence,h/1 = 0.

Thus, there exists k ∈ Σ(σB\m) such that kh = 0, then there exists a positiveinteger m such that

0 = c(h)mc(kh) = c(h)m+1c(k),

and hence,c(h)m+1 = 0.

As a consequence, h in nilpotent. Since B is reduced, B[X] is also reduced,hence, h = 0, and we have the result. Otherwise, since g1 is regular, c(g1)contains a regular element, say a1, see [14, Theorem 82]. Then we may considera maximal B(X)-sequence starting with a1, let, for instance, a1, f2, . . . , fn. Wehave

B/(a1)(X) ∼= B(X)/(a1).

By induction, we have the result. �The same result works for Gorenstein rings.

Theorem 6.8 With the same hypothesis as above, the following statementsare equivalent :

(a) A is Gorenstein with respect to P;(b) Na(A,σ) is Gorenstein.

Proof (a) =⇒ (b). This is also true as before.(b) =⇒ (a). We continue with the same notation as in the above proof.

Since B is Cohen–Macaulay, we may consider a system of parameters a1, . . . , anin B such that it is a maximal B-sequence. We have B is Gorenstein if andonly if B/(a1, . . . , an) is Gorenstein. Hence, we reduce the problem to the0-dimensional case. Therefore, the problem is to show that the 0-dimensionallocal noetherian ring B is Gorenstein whenever B(X) is Gorenstein. To showthis, we shall use a theorem [14, Theorem 221]. Let 0 �= a, b ∈ Ann(m) suchthat a �= b. Then 0 �= a/1, b/1 ∈ Ann(m(X)) and a/1 �= b/1. By the hypothesis,there exists f/g ∈ B(X) such that

(f/g)(a/1) = b/1.

As a consequence, there exists k ∈ Σ(σB\m) such that

kfa = kgb.

20 Pascual JARA

Using the properties of the content and using

c(k) = c(g) = B,

we obtainc(f)a = Bb.

Therefore, there exists x ∈ c(f) ⊆ B such that xa = b, i.e., the dimension ofAnn(m), as B/m-vector space, is one. �

If we consider regular rings instead of either Cohen–Macaulay or Gorensteinrings, the result is straightforward.

Acknowledgements This work was supported by FQM-266 (Junta de Andalucıa Research

Group) and MTM2010-20940-C02-01.

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