Localization

In this chapter we discuss the construction of rings of fractions, an operation which generalizes the construction of the rational numbers \( \mathbb{Q} \) from the ring \( \mathbb{Z} \) , or, in greater generality, the construction of the field of fractions of a domain. We will define fractions using as denominators elements of a chosen subset of the ring. We will see that, when we choose the denominators in the complement of a prime ideal, this construction produces a local ring whose maximal ideal is the extension of that prime with respect to a canonical homomorphism.