Two Applications of Nagata Rings and Modules

Abstract

Let τ be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara. First they are used to obtain the Chase's Theorem for τ-coherent rings. In particular, we obtain the w-version of the Chase's Theorem, where w is the classical star operation in ideal theory. In the second half we apply they to characterize τ-flatness in the sense of Van Oystaeyen and Verschoren.

Full text

TWO APPLICATIONS OF NAGATA RINGS AND MODULES
FANGGUI WANG AND LEI QIAO†
Abstract. Let τbe a finite type hereditary torsion theory on the category of
all modules over a commutative ring. The purpose of this paper is to give two
applications of Nagata rings and modules in the sense of Jara [15]. First they
are used to obtain the Chase’s Theorem for τ-coherent rings. In particular,
we obtain the w-version of the Chase’s Theorem, where wis the classical star
operation in ideal theory. In the second half we apply they to characterize
τ-flatness in the sense of Van Oystaeyen and Verschoren [25].
1. Introduction
Throughout the article, Rdenotes a commutative ring with nonzero identity and
the symbol Xdenotes an indeterminate over it. We use R-mod to represent the
category of all R-modules.
Recall that, for an integral domain D, the Nagata ring of Dis defined as follows:
Na(D) := D(X) := f
g|f, g ∈D[X] and c(g) = D,
where c(g) denotes the content of g. As we see in [8], this notion is essentially due
to Krull [16]. Then this ring was studied in Nagata’s book [20], using the notation
D(X), and in Samuel’s Tata volume [23] (where the notation D(X)loc was used).
The interest in D(X) is due to the fact that this ring has some strong ideal-theoretic
properties that Ditself need not have, maintaining in any case a strict relation with
the ideal structure of D, see for example [8, p. 174].
Some constructions relative to the classical Nagata ring has been studied in
particular contexts by many authors. A generalization of the classical Nagata ring
construction was considered by Kang [18, 17]. For each star operation ?of an
integral domain D, he studied the ring
f
g|f, g ∈D[X] and c(g)?=D.
We call this ring the star Nagata ring of Dand denote it by Na(D, ?). Later, in
[9], Fontana and Loper further generalized the previous construction so that, given
any semistar operation ?on an integral domain D, they define the semistar Nagata
Date: February 17, 2019.
Key Words: half-centered hereditary torsion theory; Nagata rings; Nagata modules; τ-coherent
rings; flat module; τ-flat module.
2010 Mathematics Subject Classification: 13D30; 13B30; 13F20; 13D07; 13A15.
†Corresponding Author.
1

2 F. WANG AND L. QIAO
ring of Das follows:
Na(D, ?) := f
g|f, g ∈D[X] and c(g)?=D?.
Recently, in order to extend these constructions above, Jara [15] works on a com-
mutative ring R, not necessarily an integral domain, with a half-centered hereditary
torsion theory σon R-mod, and define the Nagata ring of Rrelative to σ. He also
show that this new, and more general, Nagata ring parameterizes certain properties
(as noetherianness, Krull dimension, etc.) of the ring Rwith respect to σ.
Before reviewing Jara’s Nagata rings, we need to recall some concepts and facts
from hereditary torsion theory. For full detail on hereditary torsion theory the
reader may consult the books of Stenstr¨om [24] and Golan [12].
Let τbe a hereditary torsion theory on R-mod. Denote by Tτ(resp. Fτ) the
class of τ-torsion (resp. τ-torsionfree) R-modules. Let Mbe an R-module, and let
Nbe a submodule of M. Then we say that Nis τ-dense in Mif the quotient M/N
is τ-torsion, and τ-pure in Mif the quotient M/N is τ-torsionfree. It is known
that if pis a prime of R, then either pis τ-dense in Ror pis τ-pure in R(see [6,
Corollary 1.2]). Hence, a τ-dense ideal is not contained in any τ-pure prime ideal.
For each submodule Nof Mthere is a unique minimal τ-pure submodule N
of Mcontaining N, we call it the τ-purification of Nin M; in fact, Nsatisfies
the identity N/N =Tτ(M/N), where Tτ(M/N) is the unique maximal τ-torsion
submodule of M/N. In particular, we call the τ-purification of Min its injective
hull E(M) the τ-injective hull of Mand denote it by Eτ(M). Also, if Mis a
τ-torsionfree R-module, then the τ-injective hull of Mwill be denoted simply by
Mτ. An R-module Mis called τ-injective if M=Eτ(M), or equivalently Mis a
τ-pure submodule of E(M). For an R-module M, set Qτ(M) = Eτ(M/Tτ(M)).
Then Qτis the so-called localization functor and there is a canonical morphism
jτ:M→Qτ(M). We say that Mis τ-closed (or absolutely τ-pure) if the canonical
morphism jτ:M→Qτ(M) is an isomorphism. Clearly, an R-module Mis τ-closed
if and only if it is τ-torsionfree and τ-injective.
We represent by Lτ(R) (or simply by Lτ) the set of all τ-dense ideals of R. It
is well known that any hereditary torsion theory on R-mod is defined by the set
Lτ, as we have
Tτ={M∈R-mod |AnnR(x)∈Lτfor any x∈M}.
We represent by K(τ) the set of all τ-pure prime ideals and by C(τ) the set of all
maximal elements in K(τ).
Recall from [24] that an R-module Mis said to be Lτ-divisible if M=J M for
every ideal J∈Lτ, or equivalently, TNRM= 0 for all τ-torsion R-modules T.
It is easy to see that the class of Lτ-divisible modules is closed under epimorphic
images, direct sums and extensions.
Let Mbe an R-module. Then Mis called τ-finitely generated if Mcontains
a finitely generated τ-dense submodule; and Mis called τ-finitely presented (see
[5, 14]) if there exists an exact sequence of R-modules 0 →K→F→M→0
with Ffinitely generated free and K τ-finitely generated. Clearly, every τ-finitely
presented module is finitely generated. A ring Ris said to be τ-coherent if every
finitely generated ideal of Ris τ-finitely presented (cf. [14]).

TWO APPLICATIONS OF NAGATA RINGS AND MODULES 3
Recall that a hereditary torsion theory τon R-mod is said to be half-centered,
if for each ideal Iof Rsuch that I /∈Lτ, there exists a τ-pure prime ideal pof R
such that I⊆p(see [6, 1]). Recall also that τis said to be of finite type, if Lτhas
a cofinite subset of finitely generated ideals. It is well known that each finite type
hereditary torsion theory is half-centered.
Let pbe a prime ideal of R. Then there exists a hereditary torsion theory τR\p,
defined by pas follows:
LτR\p={I⊆R|I*p}.
As a consequence, we have the following:
(1) τR\pis a half-centered hereditary torsion theory, and
(2) a hereditary torsion theory τis half-centered if and only if it satisfies
τ=∧{τR\p|p∈K(τ)},
where the meet ∧τR\pis defined as the hereditary torsion theory such that
T∧τR\p=∩TτR\p.
Now, we recall the definition of the Nagata ring of Rrelative to a half-centered
hereditary torsion theory τfrom [15]. Let a⊆Rbe an ideal. Then we represent
by a[X] the ideal of R[X] generated by all polynomials with coefficients in a. Set
Σ(τ) = {f∈R[X]|c(f)∈Lτ}.
Then Σ(τ) is a multiplicatively closed subset of R[X] (see [15, Lemma 2.1]). Let
us represent the ring R[X]Σ(τ)simply by Na(R, τ) and call it the Nagata ring of
Rrelative to τ. For any R-module M, we define the Nagata module of Mwith
respect to τas
Na(M, τ ) = M[X]Σ(τ)= (MNRR[X])Σ(τ).
The aim of this paper is to present two applications of Jara’s Nagata rings and
modules. First, in Section 2, they are used to obtain the Chase’s Theorem for
coherent rings relative to a finite type hereditary torsion theory. In [5], Campbell
showed that if τis a finite type hereditary torsion theory and if Ris a τ-coherent
ring, then (]) the direct product of any family of Lτ-divisible flat R-modules is
flat. He also proved that if τis perfect, then Ris τ-coherent is equivalent to the
statement (]) holds. Later, in [19, Proposition 1.6], Mart´ınez Hern´andez generalized
Campbell’s results and showed that, for any hereditary torsion theory τ, if Ris τ-
coherent, then the statement (]) holds, and that the converse is true if τis of finite
type and satisfies the condition (\) that τhas an injective cogenerator Esuch that
the character module E+:= HomZ(E, Q/Z) of Eis flat. In Theorem 2.10, we
prove, in terms of Jara’s Nagata rings, that the condition (\) in [19, Proposition
1.6] is superfluous. To do so, we introduce the notion of τ-finitely presented type
modules, which generalizes the classical notion of τ-finitely presented modules. We
also show that, for a finite type hereditary torsion theory τ,Ris τ-coherent if
and only if every τ-finitely generated ideal of Ris of τ-finitely presented type (see
Theorem 2.10).
In the second half we apply them to characterize flat modules relative to a finite
type hereditary torsion theory. Recall from [25, p. 22] that an R-module Mis
said to be τ-flat if for any monomorphism u:N→N1, the kernel of the induced

4 F. WANG AND L. QIAO
morphism MNRN→MNRN1is τ-torsion. It is easy to verify that Mis τ-flat
if and only if for each morphism v:A→A1such that ker(v) is τ-torsion, we have
that the kernel of MNRA→MNRA1is τ-torsion too. In [2, Proposition 1.5],
τ-flat modules were characterized in terms of relative Tor-groups. Furthermore, if
τis a half-centered hereditary torsion theory, then an R-module Mis τ-flat if and
only if for any p∈K(τ), Mpis flat over Rp(see [21, Lemma 2.5]). In Section 3,
we show that if τis a finite type hereditary torsion theory, then an R-module Mis
τ-flat if and only if Na(M, τ ) is flat over Na(R, τ) (see Proposition 3.5).
Any undefined notions or notation is standard, as in [22, 26].
2. Chase’s Theorem for τ-coherent rings
Throughout, τdenotes a half-centered hereditary torsion theory on R-mod. We
begin with a discussion of Lτ-divisible modules. The following proposition gives
some examples of Lτ-divisible modules.
Proposition 2.1.
(1) Let p∈K(τ). Then every Rp-module is Lτ-divisible.
(2) For any Na(R, τ )-module A, there is a Na(R, τ )-isomorphism
Na(M, τ )NNa(R,τ)A∼
=MNRA.
(3) If τis of finite type, then every Na(R, τ)-module is Lτ-divisible.
Proof. (1) Assume that Mis an Rp-module and x∈M. Then for each ideal
J∈Lτ, we have J*p. Let a∈Jwith a /∈p. Hence, x=a(1
ax)∈JM. Thus,
M=JM, and so Mis Lτ-divisible.
(2)
Na(M, τ )N
Na(R,τ)
A∼
= MN
R
R[X]N
R[X]
Na(R, τ )!N
Na(R,τ)
A∼
=MN
R
A.
(3) Let Nbe a Na(R, τ )-module. Then for any τ-torsion R-module T, by (2)
and [15, Lemma 4.2], we have
TNRN∼
=Na(T, τ )NNa(R,τ)N= 0,
whence Nis Lτ-divisible as an R-module. 
Denote by Dτ(R) (or simply by Dτ) the class of Lτ-divisible flat R-modules.
One can see that if f:M→Nis an R-homomorphism such that both ker(f) and
coker(f) are τ-torsion, then for each A∈Dτ, the induce homomorphism f⊗1A:
MNRA→NNRAis isomorphic.
Now let Mbe an R-module and {Ai}be a family of R-modules over some index
set Γ. Then there is a map
θM:MN
RQ
i
Ai→Q
i
(MN
R
Ai)
given by θ(x⊗(ai)) = (x⊗ai). It is well known that θMis an epimorphism
(respectively, an isomorphism) if and only if Mis finitely generated (respectively,
finitely presented).

TWO APPLICATIONS OF NAGATA RINGS AND MODULES 5
Lemma 2.2. Let Mbe an R-module and {Ai}be a family of Lτ-divisible R-
modules over some index set Γ. Then:
(1) If Mis τ-finitely generated, then θMis an epimorphism.
(2) If Mis τ-finitely presented, then θMis an isomorphism.
Proof. (1) If Mis τ-finitely generated, then there is a finitely generated submodule
Nof Msuch that M/N is τ-torsion. Since each Aiis Lτ-divisible, we have the
following commutative diagram with exact rows.
NNR(QiAi)
θN

//MNR(QiAi)
θM

//M/N NR(QiAi)

//0
Qi(NNRAi)//Qi(MNRAi)//Qi(M/N NRAi) = 0
Note that θNis epimorphic as Nis finitely generated. Thus, it is easy to see that
θMis also an epimorphism.
(2) Let Mbe a τ-finitely presented module. Then there exists an exact sequence
of R-modules 0 →K→F→M→0 with Ffinitely generated free and K τ -finitely
generated. Consider the following commutative diagram with exact rows,
KNR(QiAi)
θK

//FNR(QiAi)
θF

//MNR(QiAi)
θM

//0
Qi(KNRAi)//Qi(FNRAi)//Qi(MNRAi)//0,
where θFis an isomorphism. By (1), θKis an epimorphism. Hence, the Five
Lemma shows that θMis an isomorphism. 
For our purpose, we called a sequence of R-modules Af
→Bg
→C τ-exact if
both ker(g)+im(f)
ker(g)and ker(g)+im(f)
im(f)are τ-torsion. It is not difficult to prove that a
sequence of R-modules A→B→Cis τ-exact if and only if for each p∈K(τ),
Ap→Bp→Cpis exact over Rp.
Now we can give a description of τ-finitely generated modules in terms of τ-exact
sequences. Indeed, one can easily see that an R-module Mis τ-finitely generated if
and only if there exists a τ-exact sequence Ff
→M→0 with Fa finitely generated
free R-module. Our next definition is motivated by this fact.
An R-module Mis said to be of τ-finitely presented type if there is a τ-exact
sequence of R-modules F1→F0→M→0 with F0, F1finitely generated free.
Clearly, every τ-finitely presented type module is τ-finitely generated. Moreover,
it is obvious that every τ-finitely presented module is of τ-finitely presented type.
But the converse is not true in general.
Example 2.3.Let Rbe a field and let D=R[X1, . . . , Xn, . . . ] be the ring of poly-
nomials in countably many variables {Xi}. Then Dis a Krull domain. Let κbe
the hereditary torsion theory defined by the height-one prime ideals of D. Then
we have that Dis a κ-Noetherian domain. Thus, the ideal m=< X1, . . . , Xn, . . . >
is κ-finitely generated. Note that for any half-centered hereditary torsion theory
τ, every τ-finitely generated module over a τ-Noetherian ring must be of τ-finitely

6 F. WANG AND L. QIAO
presented type. So mis of κ-finitely presented type. However, since mis not finitely
generated, it is not a κ-finitely presented D-module.
The following four lemmas are needed to characterize τ-finitely generated mod-
ules and τ-finitely presented type modules by τ-Nagata modules.
Lemma 2.4. Let Mbe a τ-torsionfree R-module and Na submodule of M. If
x∈E(M)such that Jx ⊆Nfor some J∈Lτ, then x∈Nτ.
Proof. Let x∈E(M) such that Jx ⊆Nfor some J∈Lτ. Then by the definition of
τ-injective hulls, we need only show that x∈E(N). To do so, we may suppose that
E(N)⊆E(M), since Nis a submodule of M. Then E(M) = E(N)LAfor some
R-module A. Now, set x=y+z, where y∈E(N) and z∈A. Therefore, for each
a∈J, we have az =ax −ay ∈E(N)TA={0}, and so az = 0. Hence, J z = 0.
But the τ-torsionfreeness of Aimplies that z= 0. Thus, x=y∈E(N). 
The next three lemmas may be simple consequences of the results in [15] (Lem-
mas 4.1 and 4.2), where is stated that, being τof finite type, the functor Na(−, τ )
is exact and vanished on τ-torsion modules. So, we omit their proofs.
Lemma 2.5. Let τbe of finite type and let Mbe a τ-torsionfree R-module. Then
Na(M, τ ) = Na(Mτ, τ ).
Lemma 2.6. Assume that τis of finite type. Let Mbe a τ-torsionfree R-module
and let A, B be submodules of M. Then Na(A, τ ) = Na(B, τ )if and only if Aτ=
Bτ.
Lemma 2.7. Let τbe of finite type and let A→B→Cbe a sequence of R-modules.
Then the sequence is τ-exact if and only if
Na(A, τ )→Na(B, τ )→Na(C, τ)
is exact over Na(R, τ ).
Proposition 2.8. Let τis of finite type and let Mbe an R-module. Then
(1) Mis τ-finitely generated if and only if Na(M, τ )is finitely generated over
Na(R, τ ).
(2) Mis of τ-finitely presented type if and only if Na(M, τ )is finitely presented
over Na(R, τ ).
Proof. (1) If Mis τ-finitely generated, then there exists a τ-exact sequence of
R-modules F→M→0 with Ffinitely generated free. Hence, by Lemma 2.7,
Na(F, τ )→Na(M, τ )→0 is exact over Na(R, τ ) and Na(F, τ ) is finitely generated
free Na(R, τ )-module. Therefore, Na(M, τ ) is finitely generated over Na(R, τ).
Conversely, assume that Na(M, τ ) is finitely generated over Na(R, τ). Then by
[15, Corollary 4.3], we may assume that Mis τ-closed. Set Na(M, τ ) = AΣ(τ),
where Ais a finitely generated R[X]-submodule of M[X]. Hence, c(A) is a finitely
generated R-module. Since A⊆c(A)[X]⊆M[X],
Na(M, τ ) = AΣ(τ)⊆Na(c(A), τ )⊆Na(M, τ ),
and so Na(c(A), τ ) = Na(M, τ ). Thus, it follows from Lemma 2.6 that M= c(A)τ.
Therefore, Mis τ-finitely generated.

TWO APPLICATIONS OF NAGATA RINGS AND MODULES 7
(2) The proof of the necessity is similar to that given in (1). For the sufficiency,
let Na(M, τ ) be finitely presented over Na(R, τ). Then by (1), we see that Mis
τ-finitely generated, and so there is a τ-exact sequence of R-modules
0→K→F→M→0
with F τ-finitely generated free. Thus, it follows from Lemma 2.7 that the sequence
0→Na(K, τ )→Na(F, τ )→Na(M, τ )→0
is exact over Na(R, τ). Hence, Na(K, τ) is a finitely generated Na(R, τ )-module,
and so Kis τ-finitely generated. Therefore, Mis of τ-finitely presented type. 
A key lemma for our main result is the following:
Lemma 2.9. Let τbe of finite type and let Mbe an R-module. If
θM:MNRNa(R, τ )Γ→(MNRNa(R, τ))Γ
is an isomorphism for any set Γ, then Mis of τ-finitely presented type.
Proof. It suffices, by Proposition 2.8, to show that Na(M, τ ) is finitely presented
over Na(R, τ). To see this, let us consider the following commutative diagram
Na(M, τ )NNa(R,τ)Na(R, τ )Γ
∼
=

θNa(M,τ )//Na(M, τ )Γ
∼
=

MNRNa(R, τ )ΓθM
∼
=
//(MNRNa(R, τ ))Γ
for any index set Γ, where the vertical isomorphisms are given by Proposition
2.1(2). Thus, θNa(M,τ )is also an isomorphism. Hence, the desired result now
follows immediately from [7, Theorem 3.2.22]. 
Recall from [3] that an R-module Mis said to be Q-flat if MΓis flat for any
index set Γ. Now we can prove the main result of this paper.
Theorem 2.10. Let τbe of finite type. Then the following statements are equiva-
lent for R.
(1) Ris a τ-coherent ring.
(2) Every τ-finitely generated ideal of Ris of τ-finitely presented type.
(3) The direct product of any family of Lτ-divisible flat R-modules is flat.
(4) Every Lτ-divisible flat R-module is Q-flat.
(5) Na(R, τ )is Q-flat over R.
Proof. (1) ⇒(2) Assume that Ris a τ-coherent ring, and that Iis a τ-finitely
generated ideal of R. Then there is a finitely generated subideal Jof Iwith I/J
τ-torsion. Let 0 →K→F→J→0 be an exact sequence of R-modules with F
finitely generated free. Since Ris τ-coherent, Jis τ-finitely presented, and so Kis
τ-finitely generated. Let F0→K→0 be a τ-exact sequence of R-modules with F0
finitely generated free. Then the sequence F0→F→I→0 is τ-exact sequence,
and so Iis of τ-finitely presented type.
(2) ⇒(1) Suppose (2) holds and let Ibe a finitely generated ideal of R. Then
there exists an exact sequence of R-modules 0 →K→F→I→0 with Ffinitely

8 F. WANG AND L. QIAO
generated free. By (2), Iis of τ-finitely presented type. Thus, it follows from
Lemma 2.7 and Proposition 2.8 that Kis τ-finitely generated, and so Iis τ-finitely
presented. Hence, Ris a τ-coherent ring.
(1) ⇒(3) See [19, Proposition 1.6].
(3) ⇒(4) This is trivial.
(4) ⇒(5) The proof follows immediately from the fact that Na(R, τ ) is Lτ-
divisible flat as an R-module (see Proposition 2.1).
(5) ⇒(2) Assume (5) holds and let Ibe a τ-finitely generated ideal of R. Then
R/I is a τ-finitely presented R-module. Since Na(R, τ ) is Q-flat over R, we have
the following commutative diagram with exact rows
0//INRNa(R, τ )Γ//
θI

Na(R, τ )Γ//R/I NRNa(R, τ)Γ//
θR/I

0
0//(INRNa(R, τ ))Γ//Na(R, τ)Γ//(R/I NRNa(R, τ ))Γ//0
for any index set Γ. By Lemma 2.2, θR/I is an isomorphism. Thus, θIis also
isomorphic, and so Iis of τ-finitely presented type by Lemma 2.9. 
A particular case of our interest is taking τto be the hereditary torsion theory
w. Following [27], an ideal Jof Ris called a Glaz-Vasconcelos ideal (a GV-ideal
for short) if Jis finitely generated and the natural homomorphism ϕ:R→J∗=
HomR(J, R) is an isomorphism. Note that the set GV(R) of GV-ideals of Ris a
multiplicative system of ideals of R. Let Mbe an R-module. Define
torGV(M) = {x∈M|J x = 0 for some J∈GV(R)}.
Thus torGV(M) is a submodule of M. Now Mis said to be GV-torsion (resp.,
GV-torsionfree) if torGV(M) = M(resp., torGV (M) = 0). Set
w= ({GV-torsion R-modules},{GV-torsionfree R-modules}).
Then it is easy to see that wis a hereditary torsion theory on R-mod. Thus, the
so-called w-modules (in the sense of [27, 26]) are exactly the w-closed modules.
Moreover, flat modules and reflexive modules are both w-closed modules. In the
integral domain case, w-closed modules were called semi-divisorial modules in [11]
and (in the ideal case) F∞-ideals in [13]. They have proved to be useful in the study
of multiplicative ideal theory and module theory. Note that
K(w) = {p∈Spec(R)|pis a prime w-closed ideal}
and
C(w) = {m∈Spec(R)|mis a maximal w-closed ideal}.
Since an R-module Mis GV-torsion if and only if Mm= 0 for each m∈C(w) (see
[26, Theorem 6.2.15]), wis a half-centered hereditary torsion theory. In fact, wis
a well-centered hereditary torsion theory (in the sense of [6]).
It is worth noting that wis also known as a classical star operation in multiplica-
tive ideal theory. The notion of coherence relative to a (semi)star operation has
been studied in several papers (see, for example, [10]). Recall from [26] that a ring
Ris said to be w-coherent if every w-finitely generated ideal of Ris of w-finitely

TWO APPLICATIONS OF NAGATA RINGS AND MODULES 9
presented type. As a corollary of Theorem 2.10, we obtain the Chase’s Theorem
for w-coherent rings.
Corollary 2.11. The following statements are equivalent for a ring R.
(1) Ris a w-coherent ring.
(2) Every finitely generated ideal of Ris w-finitely presented.
(3) The direct product of any family of Lw-divisible flat R-modules is flat.
(4) Every Lw-divisible flat R-module is Q-flat.
(5) Na(R, w)is Q-flat over R.
3. τ-flat modules
In this section, we shall give several characterizations of τ-flat modules. In
particular, when τis of finite type, we characterize τ-flat modules in terms of τ-
Nagata modules. Throughout, Fτwill denote the class of all τ-flat R-modules. For
a class Aof R-modules, we set
A>={B∈R-mod |TorR
1(A, B) = 0 for all A∈ A}.
First, we point out that vanishing of the classical Tor functor may also charac-
terize τ-flatness.
Proposition 3.1. Let p∈K(τ)and let Mbe an Rp-module. Then:
(1) Mis τ-torsionfree as an R-module.
(2) M∈ Fτ>.
(3) If M∈ Fτ, then Mis flat over R.
Proof. (1) Assume that Jx = 0 where J∈Lτ(R) and x∈M. Since J*p, we can
choose a∈Jwith a /∈p. Hence, we have x=1
a(ax) = 0. Thus, it follows that M
is a τ-torsionfree R-module.
(2) Let F∈ Fτ. Then since TorR
1(F, M ) is an Rp-module, it is τ-torsionfree as
an R-module by (1). But the τ-flatness of Fimplies that TorR
1(F, M ) is a τ-torsion
R-module. Therefore, TorR
1(F, M ) = 0, and so M∈ Fτ>.
(3) Let Nbe an R-module. Then TorR
1(M, N ) is a τ-torsion R-module as M∈
Fτ. However, it is also τ-torsionfree as an R-module, and so TorR
1(M, N ) = 0.
Hence, Mis a flat R-module. 
Proposition 3.2. The following statements are equivalent for an R-module M.
(1) Mis τ-flat.
(2) TorR
1(M, N ) = 0 for all N∈ Fτ>.
(3) TorR
1(M, N ) = 0 for all Rp-modules N, where pruns over K(τ).
Proof. (1) ⇒(2) Trivial.
(2) ⇒(3) It follows immediately from Proposition 3.1(2).
(3) ⇒(1) Assume that (3) holds, and that Nis an R-module. Then for each
p∈K(τ) we have
TorR
1(M, N )p∼
=TorRp
1(Mp, Np)∼
=TorR
1(M, Np)p= 0.
Therefore, TorR
1(M, N ) is a τ-torsion R-module, and so Mis τ-flat. 
Next, we will characterize τ-flatness by τ-Nagata modules.

10 F. WANG AND L. QIAO
Proposition 3.3. Let τbe of finite type. Then:
(1) If J∈Lτ, then Na(J, τ ) = Na(R, τ).
(2) Every Na(R, τ )-module is τ-torsionfree as an R-module.
(3) Every Na(R, τ )-module is in Fτ>.
Proof. (1) Let J∈Lτ. Then since τis of finite type, there exists a finitely generated
subideal Iof Jwith I∈Lτ. Thus, we can pick f∈R[X] with c(f) = I. Note that
f∈I[X]TΣ(τ), and so Na(R, τ ) = Na(I, τ ) = Na(J, τ).
(2) Let Nbe Na(R, τ )-module and let Jx = 0, where J∈Lτand x∈N.
Then Na(J, τ )x= 0. But Na(J, τ ) = Na(R, τ) by (1), and so x= 0. Thus, Nis
τ-torsionfree as an R-module.
(3) By using (2), this proof is similar to that of Proposition 3.1(2). 
Lemma 3.4. Assume that φ:R→Tbe a commutative ring homomorphism and
M, N be R-modules. Then:
(1) Let Bbe a T-module. Then
TorR
1(M, B )∼
=TorT
1(TNRM, B ).
(2) If Tis flat over R, then
TNRTorR
1(M, N )∼
=TorT
1(TNRM, T NRN).
(3) Let Bbe a Na(R, τ )-module. Then
TorR
1(M, N )∼
=TorNa(R,τ )
1(Na(M, τ ), N ).
(4) Na(R, τ )NRTorR
1(M, N )∼
=TorNa(R,τ )
1(Na(M, τ ),Na(N, τ )).
Proof. (1) Clearly, for any T-module Y,MNRY∼
=(TNRM)NTY. Let 0 →
K→F→B→0 be an exact sequence of T-modules with Ffree. Then since Fis
flat over R, we obtain the following commutative diagram with exact rows.
0//TorR
1(M, B )

//MRK
∼
=

//MRF
∼
=

0//TorT
1(TRM, B )//(TRM)TK//(TRM)TF
Thus, by the Five Lemma, the first vertical map is an isomorphism because the
second two are.
(2) First, note that for each R-module Y, we have
(TNRY)NT(TNRN)∼
=TNR(YNT(TNRN))
∼
=TNR((YNTT)NRN)∼
=TNR(YNRN).
Second, let 0 →A→P→M→0 be an exact sequence of R-modules with P
projective. Then the sequence
0→TorR
1(M, N )→ANRN→PNRN→MNRN→0
is exact over R. Moreover, since Tis flat over R, we obtain an exact sequence of
T-module
0→TNRA→TNRP→TNRM→0

TWO APPLICATIONS OF NAGATA RINGS AND MODULES 11
where TNRPis a projective T-module. Now the following commutative diagram
has exact rows.
0//T
R
TorR
1(M, N )

//T
R
A
R
N
∼
=

//T
R
P
R
N
∼
=

0//TorT
1T
R
M, T
R
N//T
R
A
T
T
R
N//T
R
P
T
T
R
N
Thus, (2) holds.
(3) This is a special case of (1).
(4) Since Na(R, τ ) is flat over R[X], it is also flat over R. Hence, (4) follows
immediately from (2). 
Proposition 3.5. Let τbe of finite type. Then the following statements are equiv-
alent for an R-module M.
(1) Mis τ-flat.
(2) TorR
1(M, N ) = 0 for all Na(R, τ )-modules N.
(3) Na(M, τ )is flat over Na(R, τ ).
Proof. (1) ⇒(2) By Proposition 3.3(3).
(2) ⇒(3) Assume that (2) holds. Let Nbe a Na(R, τ )-module. Then by Lemma
3.4(3), we have
TorNa(R,τ )
1(Na(M, τ ), N )∼
=TorR
1(M, N ) = 0.
Hence, Na(M, τ ) is a flat Na(R, τ )-module.
(3) ⇒(1) Let Na(M, τ ) be flat over Na(R, τ ) and Nbe an R-module. Then it
follows from Lemma 3.4(4) that
Na(TorR
1(M, N ), τ )∼
=Na(R, τ )NRTorR
1(M, N )
∼
=TorNa(R,τ )
1(Na(M, τ ),Na(N, τ )) = 0.
Thus, by [15, Lemma 4.1], we see that TorR
1(M, N ) is a τ-torsion R-module, and
so Mis τ-flat. 
Acknowledgments
The authors would like to thank the referee for several valuable comments and
suggestions, which have greatly improved this paper. The first author was partially
supported by NSFC (No. 11671283). The second author was partially supported
by NSFC (No. 11701398) and the Scientific Research Fund of Sichuan Provincial
Education Department (No. 17ZB0362).
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(Fanggui Wang) School of Mathematical Sciences, Sichuan Normal University, Chengdu,
Sichuan 610066, China
E-mail address:wangfg2004@163.com
(Lei Qiao) School of Mathematical Sciences, Sichuan Normal University, Chengdu,
Sichuan 610066, China
E-mail address:lqiao@sicnu.edu.cn