Editing Localization (commutative algebra) (section)

=== Universal property ===
The (above defined) ring homomorphism <math>j\colon R\to S^{-1}R</math> satisfies a [[universal property]] that is described below. This characterizes <math>S^{-1}R</math> up to an [[ring isomorphism|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 more technical.

The universal property satisfied by <math>j\colon R\to S^{-1}R</math> is the following: 
:If  <math>f\colon R\to T</math> is a ring homomorphism that maps every element of {{mvar|S}} to a [[unit (ring theory)|unit]] (invertible element) in {{mvar|T}}, there exists a unique ring homomorphism <math>g\colon S^{-1}R\to T</math> such that <math>f=g\circ j.</math>

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 <math>\mathcal C</math> and <math>\mathcal D</math> be the categories whose objects are [[ordered pair|pairs]] of a commutative ring and a [[submonoid]] of, respectively, the multiplicative [[monoid]] or the [[group of units]] of the ring. The [[morphism]]s of these categories are the ring homomorphisms that map the submonoid of the first object into the submonoid of the second one. Finally, let <math>\mathcal F\colon \mathcal D \to \mathcal C</math> be the forgetful functor that forgets that the elements of the second element of the pair are invertible.

Then the factorization <math>f=g\circ j</math> of the universal property defines a bijection
:<math>\hom_\mathcal C((R,S), \mathcal F(T,U))\to \hom_\mathcal D ((S^{-1}R, j(S)), (T,U)).</math>

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.