Non-Noetherian Commutative Ring Theory

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Springer Science & Business Media, Oct 31, 2000 - Mathematics - 479 pages
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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.
 

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Contents

GCD DOMAINS GAUSS LEMMA AND CONTENTS OF POLYNOMIALS
1
THE CLASS GROUP AND LOCAL CLASS GROUP OF AN INTEGRAL DOMAIN
33
MORI DOMAINS
57
WHATS NEW ABOUT INTEGERVALUED POLYNOMIALS ON A SUBSET?
75
HALFFACTORIAL DOMAINS A SURVEY
97
ON GENERALIZED LENGTHS OF FACTORIZATIONS IN DEDEKIND AND KRULL DOMAINS
117
RECENT PROGRESS ON GOINGDOWN I
139
LOCALIZING SYSTEMS AND SEMISTAR OPERATIONS
169
GENERALIZED LOCAL RINGS AND FINITE GENERATION OF POWERS OF IDEALS
287
CONNECTING TRACE PROPERTIES
313
CONSTRUCTING EXAMPLES OF INTEGRAL DOMAINS BY INTERSECTING VALUATION DOMAINS
325
EXAMPLES BUILT WITH D+M A+XBX AND OTHER FULLBACK CONSTRUCTIONS
341
TCLOSEDNESS
369
ERINGS AND RELATED STRUCTURES
387
PRIME IDEALS AND DECOMPOSITIONS OF MODULES
403
PUTTING TINVERTIBILITY TO USE
429

IDEAL THEORY IN FULLBACKS
199
COMMUTATIVE RINGS OF DIMENSION 0
229
FINITE CONDUCTOR RINGS WITH ZERO DIVISORS
251
CONSTRUCTION OF IDEAL SYSTEMS WITH NICE NOETHERIAN PROPERTIES
271
ONE HUNDRED PROBLEMS IN COMMUTATIVE RING THEORY
459
Index
477
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Page 473 - ST Chapman and WW Smith, An Analysis Using The Zaks-Skula Constant Of Element Factorizations In Dedekind Domains, J. Algebra 159 (1993), 176-190.
Page 473 - DF Anderson, ST. Chapman and WW Smith, On Krull Half-Factorial Domains With Infinite Cyclic Divisor Class Group, Houston J. Math. 20 (1994), 561-570.

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