In this case, is the largest multiplicative set such that the homomorphism is injective. The preceding example is a special case of this one. If is an element of a commutative ring and then can be identified (is canonically isomorphic to) (The proof consists of showing that this ring satisfies the above universal property.) This sort of localization plays a fundamental role in the definition of an affine scheme. If is a prime ideal of a commutative ring , the set complement of in is a multiplicative set (by the definition of a prime ideal). The ring is a local ring that is generally denoted and called the local ring of at This sort of localization is fundamental in commutative algebra, because many properties of a commutative ring can be read on its local rings. Such a property is often called a local property. For example, a ring is regular if and only if all its local rings are regular. Ring properties Localization is a rich construction that has many useful properties. In this section, only the properties relative to rings and to a single localization are considered. Properties concerning ideals, modules, or several multiplicative sets are considered in other sections. if and only if contains . The ring homomorphism is injective if and only if does not contain any zero divisors. The ring homomorphism is an epimorphism in the category of rings, that is not surjective in general. The ring is a flat -module (see for details). If is the complement of a prime ideal , then denoted is a local ring; that is, it has only one maximal ideal. Properties to be moved in another section Localization commutes with formations of finite sums, products, intersections and radicals; e.g., if denote the radical of an ideal I in R, then In particular, R is reduced if and only if its total ring of fractions is reduced. Let R be an integral domain with the field of fractions K. Then its localization at a prime ideal can be viewed as a subring of K. Moreover, where the first intersection is over all prime ideals and the second over the maximal ideals. There is a bijection between the set of prime ideals of S−1R and the set of prime ideals of R that do not intersect S. This bijection is induced by the given homomorphism R → S −1R. Saturation of