Localization of a Ring

Let R 𝑅 R be a Ring, and S ⊆ R 𝑆 𝑅 S\subseteq R a subset of R 𝑅 R . The localization of R 𝑅 R with respect to S 𝑆 S , denoted S − 1 ⁢ R superscript 𝑆 1 𝑅 S^{-1}R , is the universal way to Invert every x ∈ S 𝑥 𝑆 x\in S .

Universal property

A ring homomorphism f : R → A : 𝑓 → 𝑅 𝐴 f:R\to A inverts S 𝑆 S if the Image of S 𝑆 S under f 𝑓 f is contained in the Group of Units A × superscript 𝐴 A^{\times} . More explicitly, for every x ∈ S 𝑥 𝑆 x\in S ; f ⁢ ( x ) 𝑓 𝑥 f(x) is Invertible in A 𝐴 A .

Consider the Full Subcategory of the Coslice Category R / Ring 𝑅 Ring R/\mathrm{Ring} spanned by all S 𝑆 S -inverting maps; the localization S − 1 ⁢ R superscript 𝑆 1 𝑅 S^{-1}R is an Initial Object in this category.

More explicitly, a ring homomorphism f : R → A : 𝑓 → 𝑅 𝐴 f:R\to A factors through the localization S − 1 ⁢ R superscript 𝑆 1 𝑅 S^{-1}R if and only if f ⁢ ( S ) ⊆ A × 𝑓 𝑆 superscript 𝐴 f(S)\subseteq A^{\times} , and such factorizations are unique.

Nicer Versions

If S 𝑆 S is an Ore Set of the multiplicative monoid of R 𝑅 R , then we can represent elements of the localization with fractions instead of long multiplications of formal inverses.

Moreover, if S 𝑆 S is a Submonoid of the Center of R 𝑅 R , then we can construct the localization S − 1 ⁢ R superscript 𝑆 1 𝑅 S^{-1}R with a Commutative Localization of a Ring.