On some classes of integral domains with finitely many star operations of finite type
type: Item
instance of: scholarly article
title: On some classes of integral domains with finitely many star operations of finite type
zbMATH Open document ID: 1486.13006
author: A. Kadri
author: A. Mimouni
published in: Journal of Commutative Algebra
publication date: 2022-02-09T00:00:00Z
full work available at URL: jca.2021.13.489.full
review text: In this article, the authors investigate classes of integral domains in which there are finitely many star operations of finite type. Let \(R\) be an integral domain with quotient field \(L\), \(F(R)\) the set of nonzero fractional ideals of \(R\) and \(f(R)\) the set of nonzero finitely generated fractional ideals of \(R\). A mapping \(* : F(R) \to F(R)\), \(E \to E^*\), is called a star operation on \(R\) if the following conditions hold for all \(a \in L \setminus \{0\}\) and \(E\), \(F \in F(R)\): (1) \((aE)^* = aE^*\), (2) \(E \subseteq E^*\) and if \(E \subseteq F\) then \(E^* \subseteq F^*\), and (3) \((E^*)^*=E^*\). A star operation \(*\) is said to be of finite type (or of finite character) if for each nonzero (fractional) ideal \(E\) of \(R\), \(E^* = \cup F^*\) where the union is taken over all nonzero finitely generated subideals \(F\) of \(E\). The authors begin by investigating integral domains with at most two star operations of finite type where they are able to generalize several analogous results. Then the authors move to study Mori domains with only finitely many star operations of finite type wheer they aim to generalize certain results in the Noetherian case. The final section studies pullbacks. The paper includes several nice examples along the way to help illustrate much of the theory presented in the article. In addition to an extensive bibliography, there are several open questions/conjectures presented in the article that may be of interest.
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 7471603
zbMATH Keywords: Mori domain
zbMATH Keywords: pseudovaluation domain
zbMATH Keywords: pullbacks
zbMATH Keywords: star operation
reviewed by: Christopher P. Mooney
MaRDI profile type: Publication
OpenAlex ID: W4206028348
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DOI: 10.1216/JCA.2021.13.489
description: scientific article; zbMATH DE number 7471603