When is the dual of an ideal a ring?
type: Item
instance of: scholarly article
title: When is the dual of an ideal a ring?
zbMATH Open document ID: 0979.13006
author: Thomas G. Lucas
author: Evan Houston
author: S. Kabbaj
author: A. Mimouni
published in: Journal of Algebra
publication date: 2000-06-22T00:00:00Z
review text: Let \(R\) be a domain with quotient field \(K\). For a non-zero fractional ideal \(I\) of \(R\), the fractional ideal \(I^{-1}= (R:I)= \{x\in K \mid xI \subseteq R\}\) is called the inverse (or dual) of \(I\). -- The main results established by the authors are: (1) If \(I^{-1}\) is a ring, then \(P^{-1}\) is a ring for each minimal prime ideal \(P\) of \(I\); (2) If \(P\) is a prime ideal of \(R\), then: \(P^{-1}\) fails to be a ring \(\Leftrightarrow P\) has the form \((aR_R:b)\) and \(PR_P\) is principal; (3) The inverse of a radical ideal \(I\) is a ring \(\Leftrightarrow\) for each valuation overring \(V\) of \(R\) with \(IV \neq V\), we have \(I^{-1} \subseteq V_Q\), where \(Q\) is a prime of \(V\) which is minimal over \(IV\); (4) If \(I\) and \(J\) are radical ideals, then: \(I^{-1}\) and \(J^{-1}\) are rings \(\Leftrightarrow (I\cap J)^{-1}\) and \((I+J)^{-1}\) are rings; (5) If \(I\) is the irredundant intersection of prime ideals \(P_\alpha\), then: \(I^{-1}\) is a ring \(\Leftrightarrow\) each \(P_\alpha\) is a ring; (6) If \(I\) is an ideal in an integrally closed domain \(R\), then: \(I^{-1}\) is a ring \(\Leftrightarrow I^{-1}\subseteq V\) for each valuation overring \(V\) whose maximal ideal is minimal over \(IV\). Also the authors present examples tending to show that many of the established results are the best possible.
reviewed by: Iuliu Crivei
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 1423621
zbMATH Keywords: integral domain
zbMATH Keywords: dual ring
zbMATH Keywords: fractional ideal
MaRDI profile type: Publication
full work available at URL: jabr.1999.8142
OpenAlex ID: W2012559575
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DOI: 10.1006/JABR.1999.8142
description: scientific article; zbMATH DE number 1423621