book

Commutative Ring Theory

by Paul-Jean Cahen, Douglas L. Costa, Marco Fontana, Sarah-Eddine Kabbaj

June 2023

Intermediate to advanced

280 pages

5h 59m

English

CRC Press

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Introduction1. The Associated Sequence of Overrings2. The Main Result3. GlueingsReferences
1. Introduction2. ResultsReferences

1. Introduction2. Graded Fields3. The Ring of Polynomials Over A-Graded Field4. Criterion of Separability of Graded Fields Extension5. Case Where frac(B)/frac(A) is Normal6 Graded Field Associated with a Valuated Field7. Valuated Galois Extension8. Galois Group of KV/KTReferences
Introduction1. Chain Conditions and Associated Primes2. Mori and Non Mori Domains, Examples3. Principal Ideal ConditionsReferences
1. IntroductionCarlitz's theoremExtension to number fields2. Notations And General ResultsNotationsInclusionsLocalizations3. The Local Case4. The Case of A Discrete Valuation DomainThe inclusionsSpec (D/m D)5. GlobalizationThe case of the ring of integers of a number fieldQuadratic fieldsCyclotomic fieldsCounterexamples (Δ ≠ D)References
1. Preliminaries2. Main TheoremReferences
0. Introduction1. What Is A Radix?2. How do Radices Arise in a Natural Way?3. How can Normal Subgroups of SL(2, A) be Characterized Using Radices?4. What can be Said about Radices as Sub-Objects of Commutative Rings?5. What Remains to be Done?References
1. Introduction2. Generalized Samuel Numbers Associated with Filtrations3. The Numbers a¯f(g) And b¯f(g) Associated With Two FiltrationsReferences
References
1. The Altitude Formulaa Jaffard domains and altitude inequality formulab/ Characterization2. The Class of Domains Satisfying Residually the Altitude FormulaReferences
References
0. Introduction1. The Prime Spectrum2. Krull and valuative dimension of A+XB[X]3. Examples and Counter-ExamplesReferences
References
1. A Characterization of t-Invertibility Elements and Ideals2. Comparable Elements3. t-Invertibility and t-Local ComparabilityReferences
0 Introduction1 Locally Jaffard Rings, Af-Rings and Stably Strong S-Domains2 Pullbacks and AF-Domains3 Tensor Product of Locally Jaffard DomainsReferences
1. Introduction and Preliminary Remarks2. The ConstructionReferences
Introduction1. Divisible Modules Over Principal Domains Rings2. The Socle-Fine Class of Modules Over Specified Rings3. Injective Modules Over Semi-Artinian RingsReferences
1. Introduction2. Integrality and R[X]3. Prüfer RingsAcknowledgementsReferences
1. G-graded ringsA. Where life is easy (results which are valid for all G)B. Where one must be careful (results depend on G)2. Noetherian ℤ-graded (or ℕ-graded) rings3. Connections with Algebraic GeometryA. Projective embeddingsB. Graded modules associated with sheaves of differential formsC. The canonical ring of a curveReferences
1. Symbolic Powers, Hochster's Theorem and his Example2. Construction of Peculiar Local Domains2.0 Notation2.1 Numbering2.3 Relations3. Rees Algebras and Local Domains with Odd Non-Zero Primes3.0 Notation3.1 Numbering3.3 Relations4. Brodmann-Rotthaus Example and Ogoma's ExampleReferences
0. Preliminaries on Regular Sequences and Flat Topology1. Soundable and Expanded Modules2. Characterizations of Soundable or Expanded Subsets of a SpectrumReferences
1. t-Closed Rings2. Link With t-Closed Morphisms3. The Case of an Integral Domain4. The t-Closure of an Integral Domain5. t-Closed Rings and Quasinormal Rings6. t-Closed Overrings and SubringsReferences
Introduction1. Filtrations, Rees Rings2. Multiplicity of A Filtration in A Nœtherian Ring2.1 Multiplicity of an ideal2.2 Multiplicity of a filtration2.3 Remarks3. Integral Dependence Over A Filtration, Reduction of Filtrations3.4 Remarks3.5 Proposition (Corollary 3.6 of [6]).3.7 Proposition4. Generalized Samuel Numbers4.4 Theorem4.5 Corollary.4.6 Corollary.AcknowledgementReferences
1. Introduction2. Localization of Balanced Big Cohen-Macaulay Modules3. Permutation of Regular Sequences for Balanced Big Cohen-Macaulay ModulesReferences

Content preview from Commutative Ring Theory

23 Some Aspects of the Asymptotic Theory of Ideals. Generalization to Filtrations

DAOUD A SANGARE Département de Mathématiques, FAST, Université d’Abidjan, 22 B. P. 582, Abidjan, Cote d’Ivoire

Introduction

The asymptotic theory of ideals originated with the paper “Some asymptotic properties of powers of ideals “by Pierre SAMUEL [13]. This theory is based on the fact that in a nœtherian ring, large powers of an ideal are well behaved in some sense. It gives useful tools in commutative algebra and also in algebraic geometry. This theory has been developed at the beginning by D. Rees and M. Nagata and more recently by many other authors. It deals with topics like Samuel numbers of two ideals, homogeneous pseudo valuations, Rees valuation theorem, ...

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Publisher Resources

ISBN: 9781000949469

Commutative Ring Theory

by Paul-Jean Cahen, Douglas L. Costa, Marco Fontana, Sarah-Eddine Kabbaj

23 Some Aspects of the Asymptotic Theory of Ideals. Generalization to Filtrations

Introduction

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and much more.

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23 Some Aspects of the Asymptotic Theory of Ideals. Generalization to Filtrations

Introduction

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,and much more.

You might also like

Nanosensors, 2nd Edition

A Journey into the World of Exponential Functions

Et al

CCC (Critical Communication Capability Framework)

Publisher Resources

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.