Integral Domains in Which Every Nonzero w -Flat Ideal Is w -Invertible

: Let D be an integral domain and w be the so-called w -operation on D . We deﬁne D to be a w -FF domain if every w -ﬂat w -ideal of D is of w -ﬁnite type. This paper presents some properties of w -FF domains and related domains. Among other things, we study the w -FF property in the polynomial extension, the t -Nagata ring and the pullback construction.


Motivations and Results
Let D be an integral domain. In order to study some questions of divisibility and finite generation, Sally and Vasconcelos first studied an integral domain in which every flat ideal is finitely generated [1]. Later, El Baghdadi et al. called a domain D an FF domain if every flat ideal of D is finitely generated [2]. In the same paper, as a weaker version, the authors also defined a weakly FF property on D such that each faithfully flat ideal of D is finitely generated. It is well known that a nonzero finitely generated ideal in an integral domain is flat if and only if it is invertible. Therefore an FF domain is an integral domain in which each nonzero flat ideal is invertible, and hence a weakly FF domain can be regarded as an integral domain in which each faithfully flat ideal is invertible. In [3], Anderson and Zafrullah introduced the notion of an LPI domain; that is, an integral domain in which every nonzero locally principal ideal is invertible. Since an ideal is faithfully flat if and only if it is locally principal, the concept of weakly FF domains is precisely the same as that of LPI domains. Hence an FF domain is always an LPI domain.
Recently, in [4], the authors introduced and studied the notion of a w-LPI domain; that is, an integral domain in which every nonzero w-locally principal ideal is w-invertible. (Relevant definitions and notations will be given in Section 1.2; and for more on w-LPI domains, the readers can refer to [5].) This is, the w-operation version of LPI domains. In the view of FF domains, we can define a weakly w-FF domain; that is, an integral domain in which each w-faithfully flat w-ideal is of w-finite type. Then the notion of w-LPI domains coincides with that of weakly w-FF domains because an ideal is w-faithfully flat if and only if it is w-locally principal [6,Theorem 2.7]. Thus it is natural and reasonable to study an integral domain in which every w-flat w-ideal is of w-finite type. We call an integral domain with this property a w-FF domain. Then this is both a w-operation version of FF domains and a stronger domain than a weakly w-FF domain. As an important class of FF domains, we call D an FP domain if every flat ideal of D is principal.
In this paper, we investigate w-FF domains and some related domains. Among other things, we study the w-FF property in the polynomial extension, the t-Nagata ring and the pullback construction. More precisely, we prove that D is a w-FF domain if and only if the polynomial ring D[X] is a w-FF domain, if and only if the t-Nagata ring D[X] N v is an FF domain (Theorem 1); if K + M is t-local, then D + M is a w-FF domain if and only if D is a field and K + M is a w-FF domain, where K is a field containing D as a subring and M is a nonzero maximal ideal of K + M (Corollary 4); and the power series ring over an FP domain is an FP domain (Theorem 3). Finally, we give several examples which represent relationships among integral domains related to w-FF domains (Example 1).

Preliminaries
For the reader's better understanding, we review some definitions and notations. Let D be an integral domain with quotient field q f (D), and let F(D) be the set of nonzero fractional ideals of D. For an I ∈ F(D), set I −1 := {a ∈ q f (D) | aI ⊆ D}. The mapping on F(D) defined by I → I v := (I −1 ) −1 is called the v-operation on D; the mapping on F(D) defined by I → I t := {J v | J is a nonzero finitely generated fractional subideal of I} called the t-operation on D; and the mapping on F(D) defined by I → I w := {a ∈ q f (D) | aJ ⊆ I for some finitely generated ideal J of D such that J v = D} is called the w-operation on D. We say that an I ∈ F(D) is a v-ideal (or divisorial ideal) (respectively, t-ideal, w-ideal) if I v = I (respectively, I t = I, I w = I). It is well known that I ⊆ I w ⊆ I t ⊆ I v for all I ∈ F(D). Clearly, if an I ∈ F(D) is finitely generated, then I v = I t . Let * = t or w. Then a maximal * -ideal means a * -ideal which is maximal among proper integral * -ideals. Let * -Max(D) be the set of maximal * -ideals of D. It was shown in [7,Corollary 2.17] that the notion of maximal t-ideals coincides with that of maximal w-ideals. Additionally, it is well known that t-Max(D) = ∅ if D is not a field; and a maximal t-ideal is a prime ideal. A * -ideal I ∈ F(D) is said to be of * -finite type if I = J * for some finitely generated ideal J of D. An I ∈ F(D) is said to be * -invertible if (I I −1 ) * = D (or equivalently, I I −1 M for all maximal * -ideals M of D). If * = d, then we just say "invertible" instead of "d-invertible." It was shown that the notion of t-invertibility is precisely the same as that of w-invertibility [7, Theorem 2.18]; and an I ∈ F(D) is t-invertible if and only if I t is of t-finite type and ID M is principal for all maximal t-ideals M of D [8, Corollary 2.7]. A nonzero ideal I of D is said to be w-flat if I is t-locally flat; i.e., ID M is flat for all maximal t-ideals M of D. It was shown in [9, Corollary 3.6(4)] that an I ∈ F(D) is w-flat if and only if I w is w-flat.
Let D be an integral domain, M a D-module, and E(M) the injective envelope of M. Then the w-closure of M is defined by M w := {x ∈ E(M) | Jx ⊆ M for some finitely generated ideal J of D such that J v = D}. Let A and B be D-modules. In [6], the w-tensor product of A and B is defined as A⊗ D B := (A ⊗ D B) w . We call M a w-flat module if for every short exact sequence 0 → A → B → C → 0 of D-modules, 0 → M⊗ D A → M⊗ D B → M⊗ D C → 0 is also exact. It was shown that M is w-flat if and only if M P is a flat D P -module for every maximal w-ideal P of D.

Main Results
Throughout this section, we always assume that D is an integral domain. It is well known that an ideal I of D is flat if and only if ID P is flat for all prime ideals P of D [10, Proposition 3.10]; so a nonzero flat ideal is w-flat. Additionally, it was shown that a nonzero flat ideal is a t-ideal [11,Theorem 1.4]. Hence we have the following lemma.
where the second equality follows from the fact that a nonzero flat ideal is a t-ideal. Thus I is a t-ideal.
Our next result is the w-flat analogue of Zafrullah's result which states that a t-finite type t-ideal is flat if and only if it is invertible [12,Proposition 1]. This shows that if a w-ideal I is w-invertible but not invertible, then I is w-flat (see Lemma 2) but not flat. (⇐) Let M be a maximal t-ideal of D. Since I is w-invertible, ID M is principal; so ID M is flat. Thus I is w-flat.
We define D to be a w-FF domain if every w-flat w-ideal of D is of w-finite type (equivalently, w-invertible by Lemma 2). Clearly, if every nonzero ideal of D is a w-ideal, then D is an FF domain if and only if D is a w-FF domain. A simple example of w-FF domains is a Krull domain, because D is a Krull domain if and only if every nonzero ideal of D is w-invertible [13, Theorem 3.6]. (Recall that D is a Krull domain if there exists a family {V α } α∈Λ of rank-one essential discrete valuation overrings of D such that D = α∈Λ V α and this intersection has finite character, i.e., each nonzero nonunit in D is a nonunit in only finitely many of valuation overrings V α .) Additionally, we say that D is of finite t-character if every nonzero nonunit in D is contained in only finitely many maximal t-ideals of D, and D is a Mori domain if it satisfies the ascending chain condition on integral v-ideals. It is well known that in a Mori domain, v = t, so every t-ideal of a Mori domain is of t-finite type.  Proof. (1) Recall that a nonzero flat ideal is a t-ideal. Thus the assertion follows directly from [12, Proposition 1], because a flat ideal of a w-FF domain is of w-finite type.
(2) This is an immediate consequence of Lemma 2 and the fact that a nonzero ideal I is w-flat if and only if I w is w-flat.
(3) Let I be a w-flat w-ideal of a t-locally FF domain D, and let 0 = x ∈ I. Since D has finite t-character, there exist only finitely many maximal t-ideals of D containing x, say M 1 , . . . , M n . Fix Recall that D is a Prüfer domain (respectively, Prüfer v-multiplication domain (PvMD)) if every nonzero finitely generated ideal of D is invertible (respectively, t-invertible). It is well known that every ideal of D is flat if and only if D is a Prüfer domain [16,Theorem 4.2]. We give the PvMD version of this result. Proposition 2. The following statements are equivalent.
Since D is a PvMD, J w is w-invertible; so J w is w-flat by Lemma 2. Therefore (J w )D M is flat, and hence ID M is flat [17,Proposition 10.3]. Thus I is w-flat.
(2) ⇒ (3) ⇒ (4) These implications follow because a nonzero ideal I is w-flat if and only if I w is w-flat.
It is well known that D is a PvMD if and only if D[X] is a PvMD [8,Theorem 3.7]. Thus by Theorem 2, we obtain the following result.

Corollary 1. Every w-ideal of D is w-flat if and only if every w-ideal of D[X] is w-flat.
Recall that D is a Dedekind domain if every nonzero ideal of D is invertible. We give new characterizations of Krull domains and Dedekind domains via the (w-)FF property.  Then by Proposition 2, I w is w-flat and so I w is of w-finite type. Therefore by Lemma 2, I w , and hence, I, are w-invertible. Thus D is a Krull domain.
(2) This follows directly from (1) because D is a Dedekind domain (respectively, Prüfer domain) if and only if D is a Krull domain (respectively, PvMD) and each nonzero ideal of D is a w-ideal.
(3) Note that a valuation domain is a quasi-local Prüfer domain and that a quasi-local domain is a Dedekind domain if and only if it is a rank-one discrete valuation domain. Thus the result is an immediate consequence of (2). Next, we study the w-FF property of polynomial rings and t-Nagata rings.
We next study the w-FF property of R when T is t-local. (Recall that an integral domain is t-local if it is quasi-local whose maximal ideal is a t-ideal.) To do this, we need a simple lemma whose proof is word for word that of [2, Lemma 3.5].

Lemma 4. Given a pullback diagram ( ), if R is a w-FF domain, then D is a field.
Proof. Let d be a nonzero element of D and m be a nonzero element of M. Then m d n ∈ M for all positive integers n; so m d n ∈ R. Since { m d n R} n≥1 forms an ascending chain of principal ideals, n≥1 m d n R is a w-flat w-ideal of R. Since R is a w-FF domain, a simple modification of the proof of Corollary 3 (1) shows that n≥1 m d n R = m d k R for some integer k ≥ 1. Hence m d k+1 ∈ m d k R, which indicates that d is a unit in D. Thus D is a field.

Proof. (⇒) This was shown in Lemma 4.
(⇐) Assume that D is a field. In order to avoid the trivial case, we assume that K properly contains D. Let I be a w-flat w-ideal of R. Note that T is t-linked over R Hence we obtain Thus R is a w-FF domain.

Corollary 4. With the notation as in ( ), if T is t-local, then R is a w-FF domain if and only if D is a field and T is a (w-)FF domain.
Proof. By Lemma 4 and Theorem 2, it suffices to show that if R is a w-FF domain, then T is a w-FF domain. Since T is t-local, T is a w-FF domain if and only if T is an FF domain. Thus the result is an immediate consequence of Proposition 1(1) and [2,Corollary 3.7].
It is well known that D is a UFD if and only if every t-ideal of D is principal [13, Section 1]. Additionally, it was shown that every nonzero flat ideal is a t-ideal, and hence every flat ideal of a UFD is principal. We will say that D is an FP domain (respectively, w-FP domain) if every flat ideal (respectively, w-flat w-ideal) of D is principal. If every nonzero ideal is a w-ideal, then the notion of FP domains coincides with that of w-FP domains. Additionally, it is clear that an FP domain (respectively, w-FP domain) is an FF domain (respectively, w-FF domain). Proposition 3. The following assertions hold.
(1) A w-FP domain is an FP domain.
(2) Every invertible ideal of an FP domain is principal.
(3) Every w-invertible w-ideal of a w-FP domain is principal.

Proof.
(1) This follows directly from the fact that any nonzero flat ideal is a w-flat w-ideal.
For Propositions (2) and (3): these results come easily from the fact that an invertible (respectively, w-invertible) ideal is flat (respectively, w-flat). (2) ⇒ (3) Assume that R is an FF domain. Then D is a field [2, Corollary 3.8]; so R is t-local. Thus R is a w-FP domain.
(4) ⇒ (5) Since R is an FP domain, D is a field [2, Corollary 3.8]; so R is t-local. Hence R is a (w-)FF domain. Since V is t-local, V is a rank-one discrete valuation domain by Corollaries 2(3) and 4.
(5) ⇒ (1) Note that V is a pullback as in ( ) and is t-local. Thus the implication comes directly from Corollaries 2(3) and 4.
We give new characterizations of UFDs and PIDs in terms of the (w-)FP property.  (⇐) Let I be a nonzero ideal of D. Since D is a PvMD, I w is w-flat by Proposition 2; so I w is principal because D is a w-FP domain. Thus D is a UFD.
(2) Note that D is a PID (respectively, Prüfer domain) if and only if D is a UFD (respectively, PvMD) and each nonzero ideal of D is a w-ideal; and that if every nonzero ideal of D is a w-ideal, then D is a w-FP domain if and only if D is an FP domain. Thus the equivalence follows from (1).
We next show that the power series ring over an FP domain is an FP domain. The next examples show that any of the reverses in "UFD ⇒ FP domain ⇒ FF domain," in "w-FP domain ⇒ FP domain," in "w-FP domain ⇒ w-FF domain" and "Mori domain ⇒ w-FF domain" do not generally hold. We also give an example of a t-locally FF domain which is not a w-FF domain. (This shows that the hypothesis "finite t-character" in Proposition 1(3) is essential.). Furthermore, we construct an example of a PvMD that is not a w-FF domain. Finally, we give an example of a w-FP domain D such that D[[X]] is not a w-FP domain.  Then I is finitely generated, and hence I is invertible. Therefore I is principal because D M is local. Thus D M is an FP domain. However, since D M is not a UFD, there exists a w-invertible ideal J of D M such that J w is not principal. Therefore J w is w-flat by Lemma  We end this paper with the following two questions.

Conclusions
In this paper, we introduced the concept of a w-FF domain; that is, an integral domain in which every nonzero w-flat ideal is w-invertible. As in Figure 1, the class of w-FF domains contains Noetherian domains and Mori domains. Additionally, w-FF domains are good examples of FF domains and w-LPI domains. We extend the w-FF property to the polynomial ring, the t-Nagata ring, the composite polynomial ring and the D + M construction. Unfortunately, as in Question 1(1), we could not find an example which shows that an FF domain need not be a w-FF domain. In ensuing work, we are going to find such an example.