Lecture Notes of 14 the Unione Matematica Italiana

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1 Lecture Notes of 14 the Unione Matematica Italiana For further volumes:

2 Editorial Board Franco Brezzi (Editor in Chief) IMATI-CNR Via Ferrata 5a Pavia, Italy John M. Ball Mathematical Institute St Giles Oxford OX1 3LB United Kingdom Alberto Bressan Department of Mathematics Penn State University University Park State College PA , USA Fabrizio Catanese Mathematisches Institut Universitatstraße Bayreuth, Germany Corrado De Concini Dipartimento di Matematica Universitá di Roma La Sapienza Piazzale Aldo Moro Roma, Italy deconcini@mat.uniroma1.it Nicola Fusco Dipartimento di Matematica e Applicazioni Universitá di Napoli Federico II, via Cintia Complesso Universitario di Monte S. Angelo Napoli, Italy nfusco@unina.it Carlos E. Kenig Department of Mathematics University of Chicago 5734 University Avenue Chicago, IL USA cek@math.uchicago.edu Fulvio Ricci Scuola Normale Superiore di Pisa Piazza dei Cavalieri Pisa, Italy fricci@sns.it Gerard Van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Plantage Muidergracht TV Amsterdam, The Netherlands geer@science.uva.nl Cédric Villani Institut Henri Poincaré 11 rue Pierre et Marie Curie Paris Cedex 05 France cedric.villani@upmc.fr Persi Diaconis Department of Statistics Stanford University 450 Serra Mall Stanford, CA , USA diaconis@math.stanford.edu, cgates@stat.stanford.edu The Editorial Policy can be found at the back of the volume.

3 Marco Fontana Evan Houston Thomas Lucas Factoring Ideals in Integral Domains 123

4 Marco Fontana Università degli Studi Roma Tre Dipartimento di Matematica Rome, Italy Evan Houston Thomas Lucas University of North Carolina Mathematics and Statistics Charlotte, NC, USA ISSN ISBN ISBN (ebook) DOI / Springer Heidelberg New York Dordrecht London Library of Congress Control Number: Mathematics Subject Classification (2010): 13AXX, 13CXX, 13GXX, 14A05, 11AXX c Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface A classical generalization of the Fundamental Theorem of Arithmetic states that an integral domain is a principal ideal domain if and only if each of its proper ideals can be factored as a finite product of principal prime ideals. If the principal restriction is removed, one has a characterization of (nontrivial) Dedekind domains. The purpose of this work is to study other types of ideal factorization. Most that we consider involve writing certain types of ideals in the form J,whereJ is an ideal of some special type and is a (finite) product of prime ideals. For example, we say that a domain has weak factorization if each nondivisorial ideal can be factored as above with J the divisorial closure of the ideal and a product of maximal ideals. In a different direction, we say that a domain has pseudo-dedekind factorization if each nonzero, noninvertible ideal can be factored as above with J invertible and a product of pairwise comaximal prime ideals. In each of these cases, if the domain in question is integrally closed, then it must be a Prüfer domain. While this implies, as is often the case in multiplicative ideal theory, that Prüfer domains play an important role in our study, we do provide non-integrally closed examples for each of these types of ideal factorization. On the other hand, we also consider domains in which each proper ideal can be factored as a product of radical ideals, and such domains must be almost Dedekind (hence Prüfer) domains. This volume provides a wide-ranging survey of results on various important types of ideal factorization actively investigated by several authors in recent years, often with new or simplified proofs; it also includes many new results. It is our hope that the material contained herein will be useful to researchers and graduate students interested in commutative algebra with an emphasis on the multiplicative theory of ideals. During the preparation of this work, Marco Fontana was partially supported by a Grant MIUR-PRIN (Ministero dell Istruzione, dell Università e della Ricerca, Progetti di Ricerca di Interesse Nazionale), and Evan Houston and Thomas G. Lucas v

6 vi Preface were supported by a visiting grant from GNSAGA of INdAM (Istituto Nazionale di Alta Matematica). Rome, Italy Charlotte, North Carolina May 2012 Marco Fontana Evan Houston Thomas G. Lucas

7 Contents 1 Introduction Sharpness and Trace Properties h-local Domains Sharp and Double Sharp Domains Sharp and Antesharp Primes Trace Properties Sharp Primes and Intersections Factoring Ideals in Almost Dedekind Domains and Generalized Dedekind Domains Factoring with Radical Ideals Factoring Families for Almost Dedekind Domains Factoring Divisorial Ideals in Generalized Dedekind Domains Constructing Almost Dedekind Domains Weak, Strong and Very Strong Factorization History Weak Factorization Overrings and Weak Factorization Finite Divisorial Closure Pseudo-Dedekind and Strong Pseudo-Dedekind Factorization Pseudo-Dedekind Factorization Local Domains with Pseudo-Dedekind Factorization Strong Pseudo-Dedekind Factorization Factorization and the Ring R.X/ Factorization and Intersections of Overrings h-local Maximal Ideals Independent Pairs of Overrings vii

8 viii Contents 6.3 Jaffard Families and Matlis Partitions Factorization Examples Symbols and Definitions References Index