Modules over Principal Ideal Domains
Algebra is generous, she often gives more than is asked of her.
One of the goals of abstract algebra (and of other parts of mathematics for that matter) is to take a class of algebraic structures and show that each object in the class can be systematically constructed from simple and well-understood objects in the class. In this short chapter, we achieve this goal for the class of all finitely generated abelian groups: Each such group is isomorphic to the direct product of a finite number of cyclic groups. In fact, with little extra effort, we actually prove a more general version of this result which has far-reaching implications. This is achieved by introducing the concept of a module which, apart from its intrinsic interest, has become an indispensable tool in several areas of algebra and its applications. In the present case, the abelian groups turn out to be the modules over the ring of integers; our generalization is to look at modules over an arbitrary principal ideal domain. As a by-product, we obtain the classical description of the finitely generated abelian groups as direct products of cyclic groups.
Much of what we say about modules is motivated by abelian groups. It is customary to write abelian groups additively, and we shall do so throughout this chapter. Hence, the unity is called ...