## Non-Noetherian Commutative Ring TheoryCommutative 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 |

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### Common terms and phrases

abelian group atomic Baer ring Bezout domain characterization Cl(A Cl(R class group closure coherent ring Comm commutative ring completely integrally closed construction contained Corollary D. E. Dobbs D.D. Anderson Dedekind domain defined denote diagram of type divisorial ideals E-ring elements Example exists extension factorization finite character finite type finitely generated ideal Fontana fractional ideal GCD domain Gilmer going-down domain Heinzer Hence homomorphism i-ideal i-invertible implies integral domain integrally closed intersection invertible irreducible isomorphic Krull domain Lemma Marcel Dekker Math maximal ideal minimal prime module monoid Mori domain Noetherian domain Notes in Pure one-dimensional polynomial ring positive integer Priifer domain prime ideal principal ideal proof Proposition pullback Pure Appl PVMD quasilocal quotient field R-module residue field result ring morphism satisfies seminormal semistar operation Spec(R star operation subring surjective t-closed t-ideal Theorem theory torsion-free universally going-down valuation domain Zafrullah zero-dimensional rings

### Popular passages

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.