Introduction

In Chapters 2 and 3 in our discussion of free and injective modules, it will be assumed that the ring R has an identity 1 ≠ 0 and that all modules are unital.

One of two ways to begin the study of free modules is to define them by means of a universal property (see 2–1.7). Here the other approach is used which views free modules as a generalization of vector spaces.

It will turn out that the free modules are exactly those which are isomorphic to a direct sum of the ring R viewed as a right module. And the rank of a free module is the number of copies of R needed. Thus in the special case when R is equal to a field F, all R-modules, that is vector spaces are free, and the rank is the dimension. The analogy does not stop here. Just like in the vector space case there is a concept of a basis of a free module, and that of independence generalizes linear dependence in a vector space.

If everything easily generalized from vector spaces to free modules our whole endeavour of studying modules would become trivial, we could merely treat modules as footnotes to linear algebra. The observant reader will see here already unfolding in Chapter 2 a phenomenon which will repeat many times over and over again. Namely, many familiar vector space or commutative ring concepts either generalize, or more accurately have analogues, for arbitrary noncommutative rings but many of these are in no sense trivial generalizations of familiar patterns.