Synopses & Reviews
This volume consists of twenty-one articles by many of the most prominent researchers in non-Noetherian commutative ring theory. The articles combine in various degrees surveys of past results, recent results that have never before seen print, open problems, and an extensive bibliography. One hundred open problems supplied by the authors have been collected in the volume's concluding chapter. The entire collection provides a comprehensive survey of the development of the field over the last ten years and points to future directions of research in the area. Audience: Researchers and graduate students; the volume is an appropriate source of material for several semester-long graduate-level seminars and courses.
Synopsis
Commutative Ring Theory emerged as a distinct field of research in math- ematics only at the beginning of the twentieth century. It is rooted in nine- teenth century major works in Number Theory and Algebraic Geometry for which it provided a useful tool for proving results. From this humble origin, it flourished into a field of study in its own right of an astonishing richness and interest. Nowadays, one has to specialize in an area of this vast field in order to be able to master its wealth of results and come up with worthwhile contributions. One of the major areas of the field of Commutative Ring Theory is the study of non-Noetherian rings. The last ten years have seen a lively flurry of activity in this area, including: a large number of conferences and special sections at national and international meetings dedicated to presenting its results, an abundance of articles in scientific journals, and a substantial number of books capturing some of its topics. This rapid growth, and the occasion of the new Millennium, prompted us to embark on a project aimed at presenting an overview of the recent research in the area. With this in mind, we invited many of the most prominent researchers in Non-Noetherian Commutative Ring Theory to write expository articles representing the most recent topics of research in this area.
Table of Contents
Preface.
1. GCD Domains, Gauss' Lemma, and Contents of Polynomials;
D.D. Anderson. 2. The Class Group and Local Glass Group of an Integral Domain;
D.F. Anderson. 3. Mori Domains;
V. Barucci. 4. What's New About Integer-Valued Polynomials on a Subset?
P.-J. Cahen, J.-L. Chabert. 5. Half-Factorial Domains, a Survey;
S.T. Chapman, J. Coykendall. 6. On Generalized Lengths of Factorizations in Dedekind and Krull Domains;
S.T. Chapman, et al. 7. Recent Progress on Going-Down I;
D.E. Dobbs. 8. Localizing Systems and Semistar Operations;
M. Fontana, J.A. Huckaba. 9. Ideal Theory in Pullbacks;
S. Gabelli, E. Houston. 10. Commutative Rings of Dimension 0;
R. Gilmer. 11. Finite Conductor Rings with Zero Divisors;
S. Glaz. 12. Construction of Ideal Systems with Nice Noetherian Properties;
F. Halter-Koch. 13. Generalized Local Rings and Finite Generation of Powers of Ideals;
W. Heinzer, M. Roitman. 14. Connecting Trace Properties;
J.A. Huckaba, I. Papick. 15. Constructing Examples of Integral Domains by Intersecting Valuation Domains;
K.A. Loper. 16. Examples Built with D+M, A+XB[X] and Other Pullback Constructions;
T.G. Lucas. 17. T-Closedness;
G. Picavet, M. Picavet-L'Hermitte. 18. E-rings and Related Structures;
C. Vinsonhaler. 19. Prime Ideals and Decompositions of Modules;
R. Wiegand, S. Wiegand. 20. Putting
t-Invertibility to Use;
M. Zafrullah. 21. One Hundred Problems in Commutative Ring Theory;
S.T. Chapman, S. Glaz. Index.