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→Universal property: Corrected a minor error relevant when one attempts to localize at a multiplicative set including non-regular elements (the new property correctly handles those cases by accommodating how said non-regular elements are always mapped to 0, typically a non-unit, witnessed by the factorization of the map j through the universal quotient map of the ambient ring through the sub-ideal of its nilradical generated from the intersection of the latter and the given localizing mult... Tags: Reverted Mobile edit Mobile web edit |
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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 regular 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.
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