scholarly journals Integrally Closed Ideals and Type Sequences in One-Dimensional Local Rings

1997 ◽  
Vol 27 (4) ◽  
pp. 1065-1073 ◽  
Author(s):  
Marco D'Anna ◽  
Donatella Delfino
Keyword(s):  
Local Rings ◽  
One Dimensional ◽  
Integrally Closed ◽  
Integrally Closed Ideals

scholarly journals On the Ideal Case of a Conjecture of Huneke and Wiegand

2019 ◽  
Vol 62 (3) ◽  
pp. 847-859 ◽  
Author(s):  
Olgur Celikbas ◽  
Shiro Goto ◽  
Ryo Takahashi ◽  
Naoki Taniguchi
Keyword(s):  
Free Module ◽  
Affirmative Answer ◽  
Noetherian Rings ◽  
Ideal Case ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Integrally Closed ◽  
Integrally Closed Ideals ◽  
Torsion Free ◽  
The Ideal

AbstractA conjecture of Huneke and Wiegand claims that, over one-dimensional commutative Noetherian local domains, the tensor product of a finitely generated, non-free, torsion-free module with its algebraic dual always has torsion. Building on a beautiful result of Corso, Huneke, Katz and Vasconcelos, we prove that the conjecture is affirmative for a large class of ideals over arbitrary one-dimensional local domains. Furthermore, we study a higher-dimensional analogue of the conjecture for integrally closed ideals over Noetherian rings that are not necessarily local. We also consider a related question on the conjecture and give an affirmative answer for first syzygies of maximal Cohen–Macaulay modules.

scholarly journals $$v_c$$-Noetherian domains and Krull domains

2021 ◽  
Author(s):  
Gyu Whan Chang
Keyword(s):  
Dedekind Domain ◽  
Closed Domain ◽  
One Dimensional ◽  
Star Operation ◽  
Krull Domain ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Krull Domains

AbstractLet D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

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