Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions

✍ Scribed by Marco Fontana, Sophie Frisch, Sarah Glaz (eds.)

• Publisher: Springer-Verlag New York
• Year: 2014
• Tongue: English
• Leaves: 372
• Edition: 1
• Category: Library

✦ Synopsis

This volume presents a multi-dimensional collection of articles highlighting recent developments in commutative algebra. It also includes an extensive bibliography and lists a substantial number of open problems that point to future directions of research in the represented subfields. The contributions cover areas in commutative algebra that have flourished in the last few decades and are not yet well represented in book form. Highlighted topics and research methods include Noetherian and non- Noetherian ring theory as well as integer-valued polynomials and functions.

Specific topics include:

· Homological dimensions of Prüfer-like rings

· Quasi complete rings

· Total graphs of rings

· Properties of prime ideals over various rings

· Bases for integer-valued polynomials

· Boolean subrings

· The portable property of domains

· Probabilistic topics in Intn(D)

· Closure operations in Zariski-Riemann spaces of valuation domains

· Stability of domains

· Non-Noetherian grade

· Homotopy in integer-valued polynomials

· Localizations of global properties of rings

· Topics in integral closure

· Monoids and submonoids of domains

The book includes twenty articles written by many of the most prominent researchers in the field. Most contributions are authored by attendees of the conference in commutative algebra held at the Graz University of Technology in December 2012. There is also a small collection of invited articles authored by those who did not attend the conference. Following the model of the Graz conference, the volume contains a number of comprehensive survey articles along with related research articles featuring recent results that have not yet been published elsewhere.

✦ Table of Contents

Front Matter....Pages i-viii
Weak Global Dimension of Prüfer-Like Rings....Pages 1-23
Quasi-complete Semilocal Rings and Modules....Pages 25-37
On the Total Graph of a Ring and Its Related Graphs: A Survey....Pages 39-54
Prime Ideals in Polynomial and Power Series Rings over Noetherian Domains....Pages 55-82
Integer-Valued Polynomials: Looking for Regular Bases (A Survey)....Pages 83-111
On Boolean Subrings of Rings....Pages 113-117
On a New Class of Integral Domains with the Portable Property....Pages 119-132
The Probability That Int n ( D ) Is Free....Pages 133-151
Some Closure Operations in Zariski-Riemann Spaces of Valuation Domains: A Survey....Pages 153-173
Ten Problems on Stability of Domains....Pages 175-193
The Development of Non-Noetherian Grade and Its Applications....Pages 195-211
Stable Homotopy Theory, Formal Group Laws, and Integer-Valued Polynomials....Pages 213-223
How to Construct Huge Chains of Prime Ideals in Power Series Rings....Pages 225-238
Localizing Global Properties to Individual Maximal Ideals....Pages 239-254
Prime Ideals That Satisfy Hensel’s Lemma....Pages 255-268
Finitely Stable Rings....Pages 269-291
Integral Closure of Rings of Integer-Valued Polynomials on Algebras....Pages 293-305
On Monoids and Domains Whose Monadic Submonoids Are Krull....Pages 307-330
Integral Closure....Pages 331-351
Open Problems in Commutative Ring Theory....Pages 353-375

✦ Subjects

Commutative Rings and Algebras; Category Theory, Homological Algebra; Algebraic Topology