Chapter VIII. Commutative Rings and Their Modules

Abstract

This chapter amplifies the theory of commutative rings that was begun in Chapter IV, and it introduces modules for any ring. Emphasis is on the topic of unique factorization.

Section 1 gives many examples of rings, some commutative and some noncommutative, and introduces the notion of a module for a ring.

Sections 2–4 discuss some of the tools related to questions of factorization in integral domains. Section 2 defines the field of fractions for an integral domain and gives its universal mapping property. Section 3 defines prime and maximal ideals and relates quotients of them to integral domains and fields. Section 4 introduces principal ideal domains, which are shown to have unique factorization, and it defines Euclidean domains as a special kind of principal ideal domain for which greatest common divisors can be obtained constructively.

Section 5 proves that if $R$ is an integral domain with unique factorization, then so is the polynomial ring $R[X]$. This result is a consequence of Gauss's Lemma, which addresses what happens to the greatest common divisor of the coefficients when one multiplies two members of $R[X]$. Gauss's Lemma has several other consequences that relate factorization in $R[X]$ to factorization in $F[X]$, where $F$ is the field of fractions of $R$. Still another consequence is Eisenstein's irreducibility criterion, which gives a sufficient condition for a member of $R[X]$ to be irreducible.

Section 6 contains the theorem that every finitely generated unital module over a principal ideal domain is a direct sum of cyclic modules. The cyclic modules may be assumed to be primary in a suitable sense, and then the isomorphism types of the modules appearing in the direct-sum decomposition, together with their multiplicities, are uniquely determined. The main results transparently generalize the Fundamental Theorem for Finitely Generated Abelian Groups, and less transparently they generalize the existence and uniqueness of Jordan canonical form for square matrices with entries in an algebraically closed field.

Sections 7–11 contain foundational material related to factorization for the two subjects of algebraic number theory and algebraic geometry. Both these subjects rely heavily on the theory of commutative rings. Section 7 is a section of motivation, showing the analogy between a situation in algebraic number theory and a situation in algebraic geometry. Sections 8–10 introduce Noetherian rings, integral closures, and localizations. Section 11 uses this material to establish unique factorization of ideals for Dedekind domains, as well as some other properties.