ABSTRACT
The backbone of this book is a sequence of theorems that describe the structure of certain classes of modules. Each of these theorems explicates how the modules in question may be constructed by means of one of at most four processes: (1) choosing a submodule of R, (2) forming a factor module of R, (3) building certain extensions of these factor modules (to be explained in Chapter 7), and (4) forming coproducts. Thus R, regarded as a left module over itself, is the foundation for this book. Since this module is so important, we will go carefully through its definition again. For the underlying additive abelian group we use R with its ring addition. For the scalar multiplication of the ring R on the abelian group R, we use the ring multiplication. Reviewing the axioms of a left R-module, we have the following:
r(r′ + r″) = rr′ + rr″ by the left distributivity of ring multiplication;
40(r + r′)r″ = rr″ + r′r″ by the right distributivity of ring multiplication;
(rr′)r″ = r(r′r″) by the associativity of ring multiplication;
1r = r by the definition of the multiplicative identity of a ring.
