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=== 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 together technical, straightforward and boring.
The universal property satisfied by <math>j\colon R\to S^{-1}R</math> is the following:
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