At the beginning of Chapter 2 the comment was made that some experienced readers will have found it amazing that a whole first chapter of this book contained no mention of the concept of ideal in a commutative ring. The same experienced readers will have found it equally amazing that there has been no discussion prior to this point in the book of the concept of module over a commutative ring. Experience has indeed shown that the study of the modules over a commutative ring R can provide a great deal of information about R itself. Perhaps one reason for the value of the concept of module is that it can be viewed as putting an ideal I of R and the residue class ring R/I on the same footing. Up to now we have regarded I as a substructure of R, while R/I is a factor or ‘quotient’ structure of R: in fact, both can be regarded as R-modules.

Modules are to commutative rings what vector spaces are to fields. However, because the underlying structure of the commutative ring can be considerably more complicated and unpleasant than the structure of a field, the theory of modules is much more complicated than the theory of vector spaces: to give one example, the fact that some non-zero elements of a commutative ring may not have inverses means that we cannot expect the ideas of linear independence and linear dependence to play such a significant rôle in module theory as they do in the theory of vector spaces.

It is time we became precise and introduced the formal definition of module.