Commutative Rings with Zero Divisors

The first book-length discussion to provide a unified treatment of commutative ring theory for rings containing zero divisors by the ideal theoretic method, Commutative Rings with Zero Divisors also examines other important questions regarding the ideals of rings with zero divisors that do not have counterparts for integral domains -- for example, determining when the space of minimal prime ideals of a commutative ring is compact. Unique features of this indispensable reference/text include characterizations of the compactness of Min Spec ... development of the theory of Krull rings with zero divisors ... complete review, for rings with zero divisors, of problems on the integral closure of Noetherian rings, polynomial rings, and the ring R(X) ... theory of overrings of polynomial rings ... positive results on chained rings as homomorphic images of valuation domains ... plus much more. In addition, Commutative Rings with Zero Divisors develops properties of two important constructions for rings with zero divisors, idealization and the A + B construction. It contains a large section of examples and counterexamples as well as an index of main results. Complete with citations of the literature, this volume will serve as a reference for commutative algebraists and other mathematicians who need to know the techniques and results of the ideal theoretic method used in commutative ring theory, and as a text for graduate mathematics courses in ring theory. Book jacket.