vc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_c$$\end{document}-Noetherian domains and Krull domains

Let D be an integrally closed domain, {Vα}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{V_{\alpha }\}$$\end{document} be the set of t-linked valuation overrings of D, and vc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_c$$\end{document} be the star operation on D defined by Ivc=⋂αIVα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$\end{document} for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a vc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_c$$\end{document}-Noetherian domain if and only if D is a Krull domain, if and only if vc=v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_c = v$$\end{document} and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then vc=v\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_c = v$$\end{document} if and only if D is a Dedekind domain.

This paper consists of four sections including introduction. In Sect. 2, we review the definitions and preliminary results related to star operations. Then, in Sect. 3, we study the integrally closed domains on which v c = d, b, v, t, or w. We also give an example of integrally closed domains on which v c = d, b, w, t, and v. Finally, in Sect. 4, we prove that if D is integrally closed, then D is a v c -Noetherian domain if and only if D is a Krull domain, if and only if v c = v and each prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then v c = v if and only if D is a Dedekind domain.

Definitions related to star operations
Let D be an integral domain with quotient field K . An overring of D means a subring of K containing D. Let . For any two star operations * 1 and * 2 on D, we mean by * 1 ≤ * 2 that I * 1 ⊆ I * 2 for all I ∈ F(D). It is easy to see that if * 1 ≤ * 2 , then ( * 1 ) f ≤ ( * 2 ) f and ( * 1 ) w ≤ ( * 2 ) w . Also, * w ≤ * f ≤ * for any star operation * on D.
The most well-known examples of star operations are the d-, v-, t-, and w-operations. The d-operation is just the identity function on F(D), i.e., . We say that D is a Prüfer * -multiplication domain (P * MD) if each nonzero finitely generated ideal of D is * f -invertible. The next result is a very nice characterization of P * MDs. which is essential to the subsequent arguments of this paper.

Lemma 2.2 The following statements are equivalent for an integral domain D.
(1) D is a P * MD.
(2) * w is an e.a.b. star operation. Let * be a star operation on D. We say that D is * -Noetherian if D satisfies the ascending chain condition on integral * -ideals of D. Hence, d-Noetherian domains are just the Noetherian domains, v-Noetherian domains are Mori domains, and w-Noetherian domains are strong Mori domains. Clearly, if D is * -Noetherian, then * f = * . Also, if * 1 ≤ * 2 are star operations on D, then * 1 -Noetherian domains are * 2 -Noetherian. Hence, Noetherian domain ⇒ strong Mori domain ⇒ Mori domain.
An integral domain D is h-local if each nonzero ideal of D is contained in only finitely many maximal ideals and each nonzero prime ideal is contained in a unique maximal ideal. We say that D is an independent ring of Krull type if D is a PvMD in which each nonzero ideal of D is contained in only finitely many maximal t-ideals and each nonzero prime t-ideal is contained in a unique maximal t-ideal. It is easy to see that an h-local Prüfer domain is an independent ring of Krull type, and the converse holds if each maximal ideal is a t-ideal.

The v c -operation on integrally closed domains
Let D be an integrally closed domain. As we noted in Sect. 2, there are at least six star operations on D, say, Proof (1) ⇒ (5) Since d w = d, by Lemma 2.2, D is a PdMD, which is exactly a Prüfer domain.
(5) ⇒ (4) It is well known and easy to see that an invertible ideal is a t-ideal. Thus, the result follows.  (1) d = w.

) Every maximal ideal of D is a t-ideal.
Proof This follows directly from the fact that It is known that D is a PvMD if and only if D is integrally closed and t = w [12, Theorem 3.5]. We next give another proof of this result.

Theorem 3.3 The following statements are equivalent for an integrally closed domain D.
(1) w = t.
(3) ⇒ (1) Note that w and t are of finite character; so, it suffices to show that I w = I t for all nonzero finitely generated ideal I of D. Let I be a nonzero finitely generated ideal of D. Then, I is t-invertible, and hence, I is w-invertible, because t-Max(D) = w-Max(D). Therefore, I t = (I I −1 ) w I t ⊆ ((I I −1 ) w I t ) w = (I I −1 I t ) w = (I (I −1 I t ) w ) w = I w ⊆ I t , and thus, I w = I t .
Recall that D is a v-domain if the v-operation (equivalently, t-operation) on D is an e.a.b. star

is a PvMD and t = v. (4) D is an independent ring of Krull type in which each maximal t-ideal is t-invertible.
Proof (1) ⇒ (2) This follows, because w ≤ v c ≤ v.
(2) ⇒ (3) Clearly, t = v, because v c ≤ t ≤ v. Also, by Corollary 3.4, D is a v-domain, and since t = v, we have that D is a PvMD.
(3) ⇒ (1) By Theorem 3.3, w = t, and thus, w = v. For an explicit example, let R be the field of real numbers, Q be the algebraic closure of Q in R, y be an indeterminate over R, and R = Q + yR [y]. Then, R is an integrally closed domain which is not a v-domain [7, page 161].

v c -Noetherian domains
In and since J is finitely generated, we have x ∈ x V α ⊆ V α . Therefore, x ∈ α∈ V α = D. Thus, D is completely integrally closed.
(2) ⇒ (1) Recall that Krull domains are PvMDs with t = v. Hence, v c = v by Corollary 3.5. Note also that Krull domains are v-Noetherian. Thus, D is v c -Noetherian.
(2) ⇒ (3) It is well known that each prime t-ideal of a Krull domain is a maximal t-ideal. Thus, the result follows from Corollary 3.5.
(2) ⇔ (4) This appears in [16, Theorem 2.8]. Obviously, b-Noetherian domains are integrally closed, and hence, Noetherian domains that are not integrally closed are not b-Noetherian. However, we do not know any example of b-Noetherian domains that are not Noetherian.