This book illustrates the fundamental concepts of commutative ring theory by material on algebraic geometry and algebraic number theory and it also emphasizes the notions of a normal domain and of integral extensions. It commences with a description of how the classical Fermat's problem is related to the property of unique factorization, and shows that normal domains are the most natural class of rings suitable for the generalization of classical arithmetic. It presents a study of an ideal class group which is a measure of non-uniqueness of factorization. The book also incorporates two important results which unite and utilize earlier sections of the book's content, ie Claborn's Theorem that any abelian group is isomorphic to an ideal class group of a Dedekind domain; and a Nagata-Mori Theorem that a normalization of a noetherian domain is a Krull domain. Much of the material connects classical methods with algebraic number theory.
Highlights include a systematic exposition of geometric and number theoretical sources of commutative ring theory, the proof of finiteness of ideal class groups for rings of algebraic integers, as well as a detailed presentation of examples of rings with some special properties.