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

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.

Preface.- Weak global dimension of Prüfer-like rings (K. Adarbeh, S.- E. Kabbaj).- Quasi-complete semilocal rings and modules (D.D. Anderson).- On the total graph of a ring and its related graphs: A survey (A. Badawi).- Prime ideals in polynomials and power series rings over Noetherian domains.- Integer-valued polynomials: Looking for regular bases (J.- L. Chabert).- On Boolean subrings of rings (I. Chajda, G. Eigenthaler).- On a new class of integral domains with the portable property (D.E. Dobbs, G. Picavet, M. Picavet-L'Hermitte).- The probability that Intn(D) is free (J. Elliott).- Some closure operations in Zariski-Riemann spaces of valuation domains: A survey (C.A. Finocchiaro, M. Fontana, K.A. Loper).- Ten problems on stability of domains (S. Gabelli).- The Development of non-Noetherian grade and its applications (L. Hummel).- Stable homotopy theory, formal group laws, and integer-valued polynomials (K. Johnson).- How to construct huge chains of prime ideals in power series rings (B.G. Kang, P.T. Toan).- Localizing global properties to individual maximal idals (T.G. Lucas).- Prime ideals that satisfy Hensel's Lemma (S. McAdam).- Finitely stable rings (B. Olberding).- Integral closure of rings of integer-valued polynomials on algebras (G. Peruginelli, N.J. Werner).- On monoids and domains whose monadic submonoids are Krull (A. Reinhart).- Integral closure (I. Swanson).- Open problems in commutative ring theory (P.- J. Cahen, M. Fontana, S. Frisch, S. Glaz).