"Provides for the first time a concise introduction to general and multiplicative ideal theory, valid for commutative rings and monoids and presented in the language of ideal systems on (commutative) monoids."
Table of Contents
Part 1 General ideal theory: monoids and monoid homomorphisms; arithmetic of ideal systems; finitary and noetherian ideal systems; monoids of quotients; comparison and mappings of ideal systems; prime and primary ideals; quotients of primary ideals and primary decompositions; strictly noetherian ideal systems; the intersection theorem and the principal ideal theorem. Part 2 Multiplicative ideal theory: abstract elementary number theory; fractional divisorial ideals; invertible ideals and class groups; arithmetic of invertible and cancellative ideals; integrative closures; valuation monoids and primary monoids; ideal theory of valuation monoids; Prufer and Bezout monoids; essential homomorphisms, GCD-homomorphisms and valuations; Lorenzen monoids; quasi divisor theories; defining systems; Krull monoids and generalizations; (almost) Dedekind and Krull monoids; t-noetherian monoids; approximation theorems; divisorial defining systems and class groups; arithmetical properties of overmonoids; solutions of exercises; a guide to results on special integral domains.
"…should become the standard reference for results on ideal systems and for star operations on integral domains."