Linear independence in vectorspaces and transcendence in algebraically closed fields both are kinds of independence. In fact, they are rather simple examples of a very general notion of independence which has arisen in model theory and which is dignified by the name “nonforking”. Stability theory is concerned with classifying and investigating structures using this and derived concepts.

We begin by characterising (non-)forking in modules. Thus we give meaning to the phrases: the type q is a non-forking (=free) extension of the type p; the element a is independent from the set B over the set C. The description is in terms of the groups G(−) introduced in §2.2. Since some of the material in this and the next two chapters is used in algebraic applications, I do not assume that the reader has already encountered ideas from stability theory and so I give illustrative examples and state (and, I hope, explain) the main background theorems. All of that is in the first section.

In the examples which I mentioned above, there is a dichotomy – algebraic completely dependent)/independent – but most theories are more complicated than this, with elements exhibiting degrees of dependence on one another. In at least some cases there are ordinal ranks which measure degree of dependence: these are discussed in §2. It turns out that the rank of a type, p, is the foundation rank of the connected component of the associated group G(p) in the lattice of such connected groups. It follows that, for modules, the various stability-theoretic ranks coincide in so far as they exist (this is not true of arbitrary stable theories).