ABSTRACT
In Chapter 6, rings were introduced axiomatically (Definition 6.1, page 127), based on the examples of the integers Z, the reals R, and the ring of 2 × 2 real matrices. On the other hand, the axiomatic definition of an abstract group (Definition 4.14, page 71) was founded on the general class of concrete groups of permutations, while Cayley’s Theorem (Section 5.6) showed that each abstract group is isomorphic to a group of permutations. In this chapter, we investigate the corresponding general class of concrete rings, namely rings of endomorphisms (of abelian groups), and the accompanying concept of a module. For fields, modules are just vector spaces. For the ring of integers, modules are just abelian groups. Thus modules capture the features that are common to vector spaces and abelian groups, providing a general context for the pervasive phenomenon of linearity.
