Let R be a commutative ring containing regular elements; that is, elements a ∈ R such that a ≠ 0 and a is not a zero divisor. In this chapter we show that any commutative ring R with regular elements can be embedded in a ring Q with unity such that every regular element of R is invertible in Q. In particular, any integral domain can be embedded in a field. Indeed, by defining the general notion of ring of fractions with respect to a multiplicative subset S, we obtain a ring Rs such that there is a canonical homomorphism from R to Rs. The conditions under which a noncommutative integral domain can be embedded in a division ring are also discussed.

Rings of fractions

Definition.A nonempty subset S of a ring R is called a multiplicative set if for all s1, s2, we have s1s2 ∈ S. If in addition, each element of S is regular, then S is called a regular multiplicative set.

Clearly, the set of all regular elements is a regular multiplicative set. In particular, if R is an integral domain, then S = R – {0} is a regular multiplicative set.

Let R be a commutative ring and S a multiplicative set. Define a relation ∼ on R × S by (r,s) ∼ (r′,s′) if there exists s″ ∈ S such that s″(rs′ - r′s) = 0.