the above where is nonzero even though the fractions should be regarded as equal. The localization is a commutative ring with addition multiplication additive identity and multiplicative identity The function defines a ring homomorphism from into which is injective if and only if does not contain any zero divisors. If then is the zero ring that has as unique element. If is the set of all regular elements of (that is the elements that are not zero divisors), is called the total ring of fractions of . Universal property The (above defined) ring homomorphism satisfies a universal property that is described below. This characterizes up to an isomorphism. So all properties of localizations can be deduced from the universal property, independently from the way they have been constructed. Moreover, many important properties of localization are easily deduced from the general properties of universal properties, while their direct proof may be together technical, straightforward and boring. The universal property satisfied by is the following: If is a ring homomorphism that maps every element of to a unit (invertible element) in , there exists a unique ring homomorphism such that Using category theory, this can be expressed by saying that localization is a functor that is left adjoint to a forgetful functor. More precisely, let and be the categories whose objects are pairs of a commutative ring and a submonoid of, respectively, the multiplicative monoid or the group of units of the ring. The morphisms of these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, let be the forgetful functor that forgets that the elements of the second element of the pair are invertible. Then the factorization of the universal property defines a bijection This may seem a rather tricky way of expressing the universal property, but it is useful for showing easily many properties, by using the fact that the composition of two left adjoint functors is a left adjoint functor. Examples If is the ring of integers, and then is the field of the rational numbers. If is an integral domain, and then is the field of fractions of . The preceding example is a special case of this one. If is a commutative ring, and if is the subset of its elements that are not zero divisors, then is the total ring of fractions of .