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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 35, Num. 1 (2020), 55 – 67

doi:10.17398/2605-5686.35.1.55

Available online October 25, 2019

On the projectivity of finitely generated flat modules

A. Tarizadeh

Department of Mathematics, Faculty of Basic Sciences, University of Maragheh

P.O. Box 55136–553, Maragheh, Iran

ebulfez1978@gmail.com

Received October 21, 2018 (revised July 18, 2019) Presented by Juan B. Sancho
Accepted September 9, 2019

Abstract: In this paper, the projectivity of a finitely generated flat module of a commutative ring is

studied through its exterior powers and invariant factors and then various new results are obtained.
Specially, the related results of Endo, Vasconcelos, Wiegand, Cox-Rush and Puninski-Rothmaler on

the projectivity of finitely generated flat modules are generalized.

Key words: exterior power, invariant factor, projectivity, S-ring, specialization cone, generalization
cone.

AMS Subject Class. (2010): 13C10, 19A13, 13C11, 13E99.

1. Introduction

The main purpose of the present paper is to investigate the projectivity
of finitely generated flat modules of a commutative ring. This topic has been
the main subject of many articles in the literature over the years and it is
still of current interest, see e.g. [3, 5, 6, 7, 11, 12, 13]. Note that in general
there are finitely generated flat modules which are not projective, see Example
2.9, also see [4, Tag 00NY] as another example (note that our example is so
simple than the cited one; it is also applicable for other purposes). The main
motivation to investigate the projectivity of finitely generated flat modules
essentially originates from the fact that every finitely generated flat module
over a local ring is free, see Theorem 2.2. This result together with Theorem
2.8 play a major role in this paper.

In this paper, the projectivity of a finitely generated flat module of a
commutative ring is studied through its exterior powers and invariant factors
and then we obtain various new and interesting results. One of the features
of this study is that some major results in the literature on the projectiv-
ity of finitely generated flat modules are generalized. Specially, Theorem 3.1
generalizes [5, Theorem 1], Theorem 3.2 improves a little [12, Theorem 2.1],

ISSN: 0213-8743 (print), 2605-5686 (online)

https://doi.org/10.17398/2605-5686.35.1.55
mailto:ebulfez1978@gmail.com
https://www.eweb.unex.es/eweb/extracta/
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56 a. tarizadeh

Theorem 3.3 generalizes [13, Theorem 2], [3, Proposition 2.3], it also gen-
eralizes [11, Proposition 5.5 and Corollary 5.6] in the commutative case, and
finally Corollary 3.10 generalizes [13, Theorem 2]. In fact, Theorem 3.3 can be
viewed as a generalization of all of the above mentioned results. This theorem
is one of the main novel contributions of this paper and has many non-trivial
consequences. Theorems 3.16, 3.18 and 3.22 are another interesting results of
this paper.

For reading the present paper having a reasonable knowledge on the exte-
rior powers of a module is necessary. In this paper, all rings are commutative.

2. Preliminaries

We need the following material in the next section.

Lemma 2.1. Let M be a finitely generated R-module, let I = AnnR(M)
and let S be a multiplicative subset of R. Then S−1I = AnnS−1R(S

−1M).

Proof. It is well known and easy.

A projective R-module is also called R-projective. We also use a similar
terminology for free and flat modules.

Unlike the Kaplansky theorem [8] which states that every projective mod-
ule over a local ring is free, but this is not true for flat modules. For example,
the field of rationals Q is Zp-flat but it is not Zp-free where p is a non-zero
prime ideal of the ring of integers Z. In spite of this, in the finite case we have
the following interesting result which can be considered it as the analogue of
the Kaplansky theorem for flat modules.

Theorem 2.2. Every finitely generated flat module over a local ring is
free.

Proof. See [9, Theorem 7.10] or [4, Tag 00NZ].

Lemma 2.3. Let φ : R → S be a morphism of rings and M a finitely
generated flat R-module. Then AnnS(M ⊗R S) = AnnR(M)S.

Proof. It is a local property implied by Lemma 2.1 and Theorem 2.2.

Let M be an R-module, n ≥ 2 a natural number and let Jn be the R-
submodule of M⊗n generated by the collection of pure tensors of the form



projectivity of finitely generated flat modules 57

x1 ⊗ . . .⊗xn with xi = xj for some i 6= j. The quotient R-module M⊗n/Jn
is called the n-th exterior power of M and it is denoted by ΛnR(M) or simply
by Λn(M) if there is no confusion on the base ring R. Write Λ0(M) = R
and Λ1(M) = M. The canonical R-multilinear map η : Mn → Λn(M) given
by (x1, . . . ,xn) 7→ x1 ∧ . . . ∧ xn := x1 ⊗ . . . ⊗ xn + Jn is clearly alterna-
tive. The R-module Λn(M) together with the map η satisfy in the following
universal property. For each alternative R-multilinear map φ : Mn → N
then there exists a unique morphism of R-modules φ′ : Λn(M) → N such
that φ = φ′ ◦η.

If M is a finitely generated R-module, then Λn(M) is a finitely generated
R-module. If R → S is a morphism of rings and M is an R-module, then
Λn(M) ⊗R S as S-module is canonically isomorphic to ΛnS(M ⊗R S). It is
also well known that if M is a projective (resp. flat) R-module, then for each
natural number n, Λn(M) is a projective (resp. flat) R-module.

If M is a R-module then the n-th invariant factor of M, denoted by In(M),
is defined as the annihilator of the n-th exterior power of M, i.e., In(M) =
AnnR

(
Λn(M)

)
.

Remark 2.4. If M is a finitely generated flat R-module then Theorem 2.2
leads us to a function ψ : Spec R → N = {0, 1, 2, . . .} which is defined as
p 7→ rankRp (Mp). It is called the rank map of M. It is obvious that the rank
map is continuous if and only if it is locally constant (i.e., for each prime ideal
p of R then there exists an open neighborhood U ⊆ Spec(R) of that point
such that rankRq (Mq) = rankRp (Mp) for all q ∈ U). It is well known that
Supp

(
Λn(M)

)
= {p ∈ Spec(R) : rankRp (Mp) ≥ n}.

If φ : R → S is a morphism of rings then the induced map Spec(S) →
Spec(R) given by p 7→ φ−1(p) is denoted by φ∗.

The Jacobson radical of a ring R is denoted by J(R).
An ideal I of a ring R is called a pure ideal if the canonical ring map

R → R/I is a flat ring map. Pure ideals are quite interesting and play an
important role in commutative and non-commutative algebra (for instance,
in classifying Gelfand rings and their dual rings). An ideal I of a ring R is a
pure ideal if and only if Ip = 0 or Ip = Rp, for each prime ideal p of R. Hence,
I is an idempotent ideal.

Theorem 2.5. An ideal I of a ring R is a pure ideal if and only if Ann(f)+
I = R for all f ∈ I.

Proof. It is a local property implied by Theorem 2.2.



58 a. tarizadeh

Corollary 2.6. Let M be a finitely generated flat R-module with anni-
hilator I. Then I is a pure ideal.

Proof. It is a local property implied by Theorem 2.2 and Theorem 2.5.

Lemma 2.7. The annihilator of a finitely generated projective module is
generated by an idempotent element.

Proof. It is deduced from [2, p. 132, Proposition 3.1].

The following result is well known, see [1, Chap. II, §5.2, Théorème 1], [4,
Tag 00NX] and [12, Proposition 1.3]. As a contribution, we provide a new
proof of this result.

Theorem 2.8. Let M be a finitely generated flat R-module. Then the
following are equivalent:

(i) M is R-projective.

(ii) The invariant factors of M are finitely generated ideals.

(iii) The rank map of M is locally constant.

Proof. (i) ⇒ (ii). It is well-known that Λn(M) is a finitely generated
projective R-module and so by Lemma 2.7, In(M) is a principal ideal.

(ii) ⇒ (iii). It suffices to show that the rank map of M is Zariski con-
tinuous. By Corollary 2.6, In(M) is an idempotent ideal. Thus there ex-
ists some a ∈ In(M) such that (1 − a)In(M) = 0. Clearly a = a2 and
In(M) = Ra. By Remark 2.4, ψ

−1({n}) = Supp N ∩
(

Spec(R) \ Supp N ′
)

where N = Λn(M) and N ′ = Λn+1(M). But Supp N = Spec(R) \V (1 −a).
Moreover, Supp N ′ = V

(
In+1(M)

)
since N ′ is a finitely generated R-module.

Therefore ψ−1({n}) is an open subset of Spec R.
(iii) ⇒ (i). Apply Theorem 2.2 and [4, Tag 00NX].

In the following we give an example of a finitely generated flat module
which is not projective. It should be noted that finding such examples of
modules is not as easy as one may think at first.

Example 2.9. Let R =
∏
i≥1

A be an infinite product of copies of a non-

zero ring A and let I =
⊕
i≥1

A which is an ideal of R. If f = (fi) ∈ I,

then there exists a finite subset D of {1, 2, 3, . . .} such that fi = 0 for all



projectivity of finitely generated flat modules 59

i ∈{1, 2, 3, . . .}\D. Clearly f = fg where g = (gi) ∈ I with gi = 1 for all i ∈ D
and gi = 0 for all i ∈{1, 2, 3, . . .}\D. Hence, I is a pure ideal of R (i.e., R/I
is a finitely generated flat R-module). If R/I is R-projective then by Lemma
2.7, there exists a sequence e = (ei) ∈ R such that I = Re. Thus there exists
a finite subset E of {1, 2, 3, . . .} such that ei = 0 for all i ∈ {1, 2, 3, . . .}\E.
Clearly {1, 2, 3, . . .} \ E 6= ∅. Pick some k ∈ {1, 2, 3, . . .}\ E. There is some
r = (ri) ∈ R such that (δi,k)i≥1 = re where δi,k is the Kronecker delta. In
particular, 1A = rkek = rk0A = 0A. This is a contradiction. Therefore R/I is
not R-projective.

3. Projectivity: main results

Throughout this section, M is a finitely generated flat R-module.

The following technical result generalizes [5, Theorem 1].

Theorem 3.1. Let R → S be an injective ring map. Then M is R-pro-
jective if and only if M ⊗R S is S-projective.

Proof. Let M ⊗R S be S-projective. Without loss of generality, we may
assume that R ⊆ S is an extension of rings. First we shall prove that I =
AnnR(M) is a principal ideal. By Lemma 2.3, IS = L where L = AnnS(N)
and N = M ⊗R S. Hence by Lemma 2.7, there is an idempotent e ∈ S such
that IS = Se. Let J = S(1 − e) ∩R. Clearly IJ = 0. We have I + J = R.
If not, then there exists a prime ideal p of R such that I + J ⊆ p. Thus, by
Theorem 2.2, Ip = 0. Therefore the extension of IS under the canonical map
S → S⊗R Rp is zero. Thus there exists an element s ∈ R\p such that se = 0
and so s = s(1 − e). Hence s ∈ J. But this is a contradiction. Therefore
I + J = R. It follows that there is an element c ∈ I such that c = c2 and
I = Rc. Now, let n ≥ 1. We have Λn(M) is a finitely generated flat R-module.
Moreover, Λn(M) ⊗R S is S-projective because it is canonically isomorphic
to ΛnS(M ⊗R S). Thus, by what we have proved above, In(M) is a principal
ideal. Hence, by Theorem 2.8, M is R-projective. The reverse is easy and
well known.

The following result is also technical and generalizes [12, Theorem 2.1].

Theorem 3.2. Let J be an ideal of R which is contained in the Jacobson
radical of R. If M/JM is R/J-projective, then M is R-projective.



60 a. tarizadeh

Proof. First we shall prove that I = AnnR(M) is a principal ideal. By
Lemma 2.3, L = I + J where L = AnnR(M/JM). Also, by Lemma 2.7,
AnnR/J(M/JM) = L/J is a principal ideal. This implies that I = Rx + I∩J
for some x ∈ R since L/J = (I + J)/J is canonically isomorphic to I/(I ∩J).
I = Rx: Let m be a maximal ideal of R. By Theorem 2.2, Im is either the
whole localization or the zero ideal. If Im = 0 then (Rx)m = 0 since Rx ⊆ I.
If Im = Rm then I is not contained in m. Thus Rx is also not contained in
m since I ∩ J ⊆ J ⊆ m. Hence (Rx)m = Rm. Therefore I = Rx. Now let
n ≥ 1 and let N = Λn(M). Then N/JN is R/J-projective, because N/JN as
R/J-module is canonically isomorphic to Λn

R/J
(M/JM) and Λn

R/J
(M/JM) is

R/J-projective. But N is a finitely generated flat R-module. Therefore, by
what we have proved above, In(M) = AnnR(N) is a principal ideal. Thus the
invariant factors of M are finitely generated ideals and so by Theorem 2.8, M
is R-projective.

Motivated by the Grothendieck’s relative point of view, then we obtain
the following result which (beside Theorems 2.2 and 2.8) is one of the most
powerful results on the projectivity of finitely generated flat modules.

Theorem 3.3. Let φ : R → S be a ring map whose kernel is contained in
the Jacobson radical of R. Then M is R-projective if and only if M ⊗R S is
S-projective.

Proof. Let M⊗R S be S-projective. Clearly M/JM is a finitely generated
flat R/J-module and M/JM ⊗R/J S ' M ⊗R S is S-projective where J =
Ker φ. Moreover R/J can be viewed as a subring of S via φ. Therefore, by
Theorem 3.1, M/JM is R/J-projective. Then by applying Theorem 3.2, we
get that M is R-projective. The reverse is easy and well known.

The above theorem has many consequences.

Recall that a ring R is called an S-ring (“S” referes to Sakhajev) if every
finitely generated flat R-module is R-projective.

Corollary 3.4. Let φ : R → S be a ring map whose kernel is contained
in the Jacobson radical of R. If S is an S-ring then R is as well.

Proof. If M is a finitely generated flat R-module then M ⊗R S is a finitely
generated flat S-module and so, by the hypothesis, it is S-projective. There-
fore by Theorem 3.3, M is R-projective.



projectivity of finitely generated flat modules 61

Corollary 3.5. If there exists a prime ideal p of a ring R such that the
kernel of the canonical ring map R → Rp is contained in the Jacobson radical
of R, then R is an S-ring.

Proof. By Theorem 2.2, every local ring is an S-ring, then apply Corol-
lary 3.4.

Corollary 3.6. If there exists a prime ideal p of a ring R such that the
canonical ring map R → Rp is injective, then R is an S-ring.

Proof. It is an immediate consequence of Corollary 3.5.

Corollary 3.7. Every integral domain is an S-ring.

Proof. It is an immediate consequence of Corollary 3.6.

Corollary 3.8. If the Jacobson radical of a ring R contains a prime ideal
p of R, then R is an S-ring.

Proof. Clearly Ker π ⊆ p where π : R → Rp is the canonical ring map.
Therefore by Corollary 3.5, R is an S-ring.

Another proof. By Corollary 3.7, R/p is an S-ring. Thus by Corollary 3.4,
R is an S-ring.

Remark 3.9. Let S be a subset of a ring R. The polynomial ring R[xs : s ∈
S] modulo I is denoted by S(−1)R where the ideal I is generated by elements
of the form sx2s −xs and s2xs − s with s ∈ S. We call S(−1)R the pointwise
localization of R with respect to S. Amongst them, the pointwise localization
of R with respect to itself, namely R(−1)R, has more interesting properties; for
further information please consult with [10]. Note that Wiegand [13] utilizes
the notation R̂ instead of R(−1)R. Clearly η(s) = η(s)2(xs + I) and xs +
I = η(s)(xs + I)

2 where η : R → S(−1)R is the canonical map and the pair
(S(−1)R,η) satisfies in the following universal property: “for each such pair
(A,φ), i.e. φ : R → A is a ring map and for each s ∈ S there is some
c ∈ A such that φ(s) = φ(s)2c and c = φ(s)c2, then there exists a unique
ring map ψ : S(−1)R → A such that φ = ψ ◦η”. Now let p be a prime ideal
of R and consider the canonical map π : R → κ(p) where κ(p) is the residue
field of R at p. By the above universal property, there is a (unique) ring
map ψ : S(−1)R → κ(p) such that π = ψ ◦ η. Thus η induces a surjection
between the corresponding spectra. This, in particular, implies that the kernel



62 a. tarizadeh

of η is contained in the nilradical of R. Using this, then the following result
generalizes [13, Theorem 2].

Corollary 3.10. If there exists a subset S of R such that M ⊗R S(−1)R
is S(−1)R-projective, then M is R-projective.

Proof. It is an immediate consequence of Theorem 3.3.

In what follows we get some new results essentially based on the referee’s
excellent comments.

Corollary 3.11. Let φ : R → S be a morphism of rings such that the
induced map φ∗ has the dense image. Then M is R-projective iff M ⊗R S is
S-projective.

Proof. It is an immediate consequence of Theorem 3.3, because from equal-
ity Im φ∗ = Spec(R) we get that the kernel of φ is contained in J(R), the
Jacobson radical of R.

Lemma 3.12. Let
{
Rk,mk

}
k

be a family of local rings. Then the kernel
of the canonical ring map π : R =

∏
k

Rk →
∏
k

Rk/mk is the Jacobson radical

of R.

Proof. If the sequence x = (xk) ∈ R is a member of Ker π then xk ∈ mk
for all k. To prove x ∈ J(R) it suffices to show 1 + xy is invertible in R
for all y = (yk) ∈ R. For each k, there exists some zk ∈ Rk such that
(1 + xkyk)zk = 1 because Rk is a local ring. It follows that (1 + xy)z = 1
where z = (zk). Conversely, let x ∈ J(R). For each k, then Mk := π−1k (mk)
is a maximal ideal of R because the ring map R/Mk → Rk/mk induced by
the canonical projection πk : R → Rk is an isomorphism. Therefore x ∈ Mk
for all k.

Corollary 3.13. Let X ⊆ Spec(R) be a subset. Then the following
statements are equivalent:

(i) M ⊗R S is S-projective where S =
∏
p∈X

κ(p).

(ii) M ⊗R S′ is S′-projective where S′ =
∏
p∈X

R/p.

(iii) M ⊗R S′′ is S′′-projective where S′′ =
∏
p∈X

Rp.



projectivity of finitely generated flat modules 63

If moreover
⋂

p∈X
p ⊆ J(R), then the above statements are equivalent with the

following:

(iv) M is R-projective.

Proof. It is an immediate consequence of Theorem 3.3.

The subsets Min(R) and Max(R) are typical examples which satisfy the
hypothesis of Corollary 3.13.

Corollary 3.14. Consider the following commutative diagram of rings

R //

��

S′

��
S

φ // T

in which the kernel of φ is contained in the Jacobson radical of S. If M ⊗R S′
is S′-projective, then M ⊗R S is S-projective.

Proof. If M ⊗R S′ is S′-projective then it is easy to see that

(M ⊗R S′) ⊗S′ T ' M ⊗R T ' (M ⊗R S) ⊗S T

is T-projective. But M ⊗R S is a finitely generated flat S-module. Therefore
by Theorem 3.3, M ⊗R S is S-projective.

Definition 3.15. If X is a subset of Spec(R) then we call
⋃

p∈X
V (p) the

specialization cone of X and it is denoted by Xs. Dually, we call
⋃

p∈X
Λ(p)

the generalization cone of X and it is denoted by Xg where Λ(p) = {q ∈
Spec(R) : q ⊆ p}.

Theorem 3.16. Let X ⊆ Spec(R) be a subset. Put S :=
∏
p∈X

κ(p) and

S′ :=
∏

p∈Xs
κ(p). Then M ⊗R S is S-projective iff M ⊗R S′ is S′-projective.

Proof. Consider the canonical injective ring map T =
∏
p∈X

R/p → S. Then

by Theorem 3.1, M ⊗R S ' (M ⊗R T) ⊗T S is S-projective iff M ⊗R T is
T-projective. By the axiom of choice, we obtain a function σ : Xs → X such



64 a. tarizadeh

that σ(p) ⊆ p for all p ∈ Xs and σ(p) = p for all p ∈ X. For each p ∈ X,
consider the canonical injective ring map R/p →

∏
q∈σ−1(p)

R/q. Then we get

the canonical injective ring map T → T ′ =
∏

p∈Xs
R/p. Again by Theorem

3.1, M ⊗R T is T-projective iff M ⊗R T ′ is T ′-projective. Similarly above, by
applying Theorem 3.1 to the canonical injective ring map T ′ → S′, we get
that M ⊗R T ′ is T ′-projective iff M ⊗R S′ is S′-projective.

Corollary 3.17. Let p be a prime ideal of a ring R and put S :=∏
q∈V (p)

κ(q). Then M ⊗R S is S-projective.

Proof. It is an immediate consequence of Theorem 3.16.

Theorem 3.18. Let X ⊆ Spec(R) be a subset. Put S :=
∏
p∈X

κ(p) and

S′ :=
∏

p∈Xg
κ(p). Then M ⊗R S is S-projective iff M ⊗R S′ is S′-projective.

Proof. The kernel of the canonical ring map T =
∏
p∈X

Rp → S is the

Jacobson radical of T , see Lemma 3.12. Therefore by Theorem 3.3, M⊗RS '
(M ⊗R T) ⊗T S is S-projective if and only if M ⊗R T is T-projective. By the
axiom of choice, there exists a function σ : Xg → X such that p ⊆ σ(p)
for all p ∈ Xg and σ(p) = p for all p ∈ X. For each p ∈ X, consider the
canonical injective ring map Rp →

∏
q∈σ−1(p)

Rq. Then we get the canonical

injective ring map T → T ′ =
∏

p∈Xg
Rp. Thus by Theorem 3.1, M ⊗R T is

T-projective iff M ⊗R T ′ is T ′-projective. Again by Lemma 3.12, the kernel
of the canonical ring map T ′ → S′ is contained in the Jacobson radical of T ′.
Hence by Theorem 3.3, M ⊗R T ′ is T ′-projective if and only if M ⊗R S′ is
S′-projective.

Corollary 3.19. Let p be a prime ideal of a ring R and put S :=∏
q∈Λ(p)

κ(q). Then M ⊗R S is S-projective.

Proof. It is an immediate consequence of Theorem 3.18.



projectivity of finitely generated flat modules 65

Corollary 3.20. Let X ⊆ Y ⊆ Spec(R) be two subsets such that either
Y ⊆ Xs or Y ⊆ Xg. Put S :=

∏
p∈X

κ(p) and S′ :=
∏
p∈Y

κ(p). Then M ⊗R S is

S-projective iff M ⊗R S′ is S′-projective.

Proof. If Y ⊆ Xs then Xs = Ys, so apply Theorem 3.16 in this case. But
if Y ⊆ Xg then Xg = Yg, and so apply Theorem 3.18.

Lemma 3.21. Let φ : R → S be a morphism of rings and put S′ :=∏
p∈Im φ∗

κ(p). Then M ⊗R S is S-projective if and only if M ⊗R S′ is S′-projec-

tive.

Proof. If p ∈ Im φ∗ and q ∈ (φ∗)−1(p) then we have the canonical ring
map κ(p) → κ(q) which is injective since every ring map from a field into
a non-zero ring is injective. Then we get the canonical injective ring map
κ(p) →

∏
q∈(φ∗)−1(p)

κ(q). So we get the canonical injective ring map S′ → T =∏
q∈Spec(S)

κ(q) which fits in the following commutative diagram

R
φ //

��

S

��
S′ // T

where the unnamed arrows are the canonical morphisms. It is easy to see
that the kernel of the canonical morphism S → T is the nilradical of S which
is contained in the Jacobson radical of S. Then the assertion is deduced by
twice using of Corollary 3.14.

Given a subset X ⊆ Spec(R), denote X(1) := (Xs)g and X(1) := (Xg)s, and
inductively X(n) := (X(n−1))(1) and X(n) := (X(n−1))(1). Note that in general,

X(n) 6= X(n). For example, if X = {2Z} ⊆ Spec(Z) then X(1) = {0, 2Z} but
X(1) = Spec(Z).

Theorem 3.22. Let φ : R → S be a morphism of rings and X = Im φ∗.
Assume there exists some n ≥ 1 such that

⋂
p∈X(n)

p is contained in the Jacobson

radical of R. If M ⊗R S is S-projective then M is R-projective.



66 a. tarizadeh

Proof. By Lemma 3.21, M ⊗R S is S-projective iff M ⊗R S′ is S′- projec-
tive where S′ :=

∏
p∈X

κ(p). By the successive applications of Theorems 3.16

and 3.18, eventually after finite times we get that M ⊗R S′ is S′-projective
iff M ⊗R T is T-projective where T =

∏
p∈X(n)

κ(p). But the kernel of the

canonical ring map R → T is equal to
⋂

p∈X(n)
p. Thus by Theorem 3.3, M is

R-projective.

Theorem 3.23. Let φ : R → S be a morphism of rings and X = Im φ∗.
Suppose there exists some n ≥ 1 such that

⋂
p∈X(n)

p is contained in the Jacobson

radical of R. If M ⊗R S is S-projective, then M is R-projective.

Proof. It is proven exactly like Theorem 3.22.

Acknowledgements

The author would like to give sincere thanks to the referee for very
careful reading of the paper and for his/her very valuable and excellent
suggestions and comments which greatly improved the paper. I would
also like to thank Professors Wolmer Vasconcelos and Jesús M.F. Castillo
for various scientific correspondences during the writing of this paper.

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projectivity of finitely generated flat modules 67

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	Introduction
	Preliminaries
	Projectivity: main results