Fields, especially skew fields, are generally constructed as the field of fractions of some ring, but of course not every ring has a field of fractions and for a given ring it may be quite difficult to decide if a field of fractions exists. While a full discussion of this question is left to Ch. 4, for the moment we shall bring some general observations on the kind of conditions to expect (mainly quasi-identities) in 1.2 and give some necessary conditions relating to the rank of free modules in 1.4, as well as some sufficient conditions. On the one hand there is the Ore condition in 1.3, generalizing the commutative case; on the other hand and perhaps less familiar, we have the trivializability of relations, leading to semifirs in 1.6, which include free algebras and coproducts of fields, as we shall see in Ch. 5. Some general relations between matrices over rings, and the applications to the factorization of elements over principal ideal domains (needed later) are described in 1.5.

Although readers will have met fields before, a formal definition is given in 1.1 and is contrasted there with the definition of near fields, which however will not occupy us further. The final section 1.7 deals with the matrix functor and its left adjoint, the matrix reduction functor, which will be of use later in constructing counter-examples.