Progress in Commutative Algebra 2: Closures, Finiteness and Factorization

Open Access Published by De Gruyter 2012

Progress in Commutative Algebra 2

Closures, Finiteness and Factorization

Edited by: Christopher Francisco , Lee C. Klingler , Sean M. Sather-Wagstaff and Janet C. Vassilev
With contributions by: Jason G. Boynton , Ela Celikbas , Scott T. Chapman , Jim Coykendall , Florian Enescu , Neil Epstein , Christina Eubanks-Turner , Sarah Glaz , Ulrich Krause , Bruce Olberding , Sean Sather-Wagstaff , Ryan Schwarz , Karl Schwede , Laura Sheppardson , Sandra Spiroff , Kevin Tucker and John J. Watkins

In the series De Gruyter Proceedings in Mathematics

https://doi.org/10.1515/9783110278606

Overview
Contents

About this book

This is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry).

This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and one by Enescu which discusses the action of the Frobenius on finite dimensional vector spaces both of which are related to tight closure.

Finiteness properties of rings and modules or the lack of them come up in all aspects of commutative algebra. However, in the study of non-noetherian rings it is much easier to find a ring having a finite number of prime ideals. The editors have included papers by Boynton and Sather-Wagstaff and by Watkins that discuss the relationship of rings with finite Krull dimension and their finite extensions. Finiteness properties in commutative group rings are discussed in Glaz and Schwarz's paper. And Olberding's selection presents us with constructions that produce rings whose integral closure in their field of fractions is not finitely generated. The final three papers in this volume investigate factorization in a broad sense. The first paper by Celikbas and Eubanks-Turner discusses the partially ordered set of prime ideals of the projective line over the integers. The editors have also included a paper on zero divisor graphs by Coykendall, Sather-Wagstaff, Sheppardson and Spiroff. The final paper, by Chapman and Krause, concerns non-unique factorization.

Author / Editor information

Christopher Francisco, Oklahoma State University, Stillwater, Oklahoma, USA; Lee C. Klingler, Florida Atlantic University, Boca Raton, Florida, USA; Sean M. Sather-Wagstaff, North Dakota State University, Fargo, North Dakota, USA; Janet Vassilev, University of New Mexico, Albuquerque, New Mexico, USA.

Supplementary Materials

Contents

Topics

Frontmatter Open Access Download PDF	i
Preface Open Access Download PDF	v
Contents Open Access Download PDF	vii
A Guide to Closure Operations in Commutative Algebra Neil Epstein Open Access Download PDF	1
A Survey of Test Ideals Karl Schwede and Kevin Tucker Open Access Download PDF	39
Finite-dimensional Vector Spaces with Frobenius Action Florian Enescu Open Access Download PDF	101
Finiteness and Homological Conditions in Commutative Group Rings Sarah Glaz and Ryan Schwarz Open Access Download PDF	129
Regular Pullbacks Jason G. Boynton and Sean Sather-Wagstaff Open Access Download PDF	145
Noetherian Rings without Finite Normalization Bruce Olberding Open Access Download PDF	171
Krull Dimension of Polynomial and Power Series Rings John J. Watkins Open Access Download PDF	205
The Projective Line over the Integers Ela Celikbas and Christina Eubanks-Turner Open Access Download PDF	221
On Zero Divisor Graphs Jim Coykendall, Sean Sather-Wagstaff, Laura Sheppardson and Sandra Spiroff Open Access Download PDF	241
A Closer Look at Non-Unique Factorization via Atomic Decay and Strong Atoms Scott T. Chapman and Ulrich Krause Open Access Download PDF	301

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Graduate and PhD Students, Researchers, Lecturers; Academic Libraries
Pages
- Front matter: 10
- Main content: 315
Keywords: Commutative Algebra; Closure; Decomposition; Factorization

eBook

Published: April 26, 2012
ISBN: 9783110278606

Hardcover

Published: April 26, 2012
ISBN: 9783110278590