Let be an integral domain and let be a set of denominators. Prove that is isomorphic to a subring of the field of fractions of . (In particular, is an integral domain.)

We begin with a lemma.

Lemma: Let be a commutative ring. Let be sets of denominators in with . Then the map given by is an injective ring homomorphism. Proof: (Well-defined) Suppose in . Then in , and thus in . Thus is well-defined. (Preserves sums) Note that . (Preserves products) Similar to the sum-preservation proof. (Injective) Suppose . Then , so that in .

The present exercise follows because every set of denominators is contained in . Because the field of fractions of is a field, any subring is an integral domain.