Notation. Λ, Λ1, Λ2, etc., denote rings, which need not be commutative, each having an identity element. Z denotes the ring of integers.

Abstract mappings

From now on we shall be encountering a particular type of phenomenon with such frequency that it will be worth our while to introduce a special terminology in order to describe it. This is achieved by introducing the concepts of category and functor.

In dealing with modules we are, above all, concerned with certain objects, namely, the modules themselves, and certain natural mappings of these objects, to wit, their homomorphisms. Roughly speaking, and we shall make this idea precise very shortly, a collection of ‘objects’ together with certain ‘natural mappings’ of these objects into one another constitute what is called a category. Modules (over a fixed ring) and their homomorphisms form one category, and, as a contrasting example, we may mention the category formed by topological spaces and continuous mappings. However, the concept is a very comprehensive one indeed and examples could be multiplied indefinitely.

Suppose now that with each object of a category is associated an object of some other category and that, in some way, a mapping of objects in the first category gives rise to a mapping of the corresponding objects in the second category. This situation, again speaking very informally, is described by saying that we have a functor from the first to the second category or that we have a, functor of a single variable.