This chapter studies ways of embedding rings in fields and more generally, the homomorphisms of rings into fields. For a commutative ring such homomorphisms can be described completely in terms of prime ideals, and we shall see that a similar, but less obvious, description applies to quite general rings.
After some generalities on the rings of fractions obtained by inverting matrices (Section 7.1) and on R-fields and their specializations (Section 7.2), we introduce in Section 7.3 the notion of a matrix ideal. This corresponds to the concept of an ideal in a commutative ring, but has no direct interpretation. The analogue of a prime ideal, the prime matrix ideal, has properties corresponding closely to those of prime ideals, and in Section 7.4 we shall see that the prime matrix ideals can be used to describe homomorphisms of general rings into fields, just as prime ideals are used in the commutative case. This follows from Theorem 4.3, which characterizes prime matrix ideals as ‘singular kernels’, i.e. the sets of matrices that become singular under a homomorphism into some field.
This characterization is applied in section 7.5 to derive criteria for a general ring to be embeddable in a field, or to have a universal field of fractions. These results are used to show that every Sylvester domain (in particular every semifir) has a universal field of fractions.