Localization and local rings

Let S be a multiplicatively closed subset of a commutative unital ring R containing one. Fractions r/s and r′/s′ are identified if some t in S kills rs′−r′s. The extra multiplier is essential when R has zero divisors. In particular, r/1 is zero precisely when some element of S annihilates r. If zero belongs to S, the localization is the zero ring; otherwise it can still kill nonzero zero divisors. For a module M the same rule gives m/s, and m/1 vanishes exactly when a denominator-set element kills m.

The universal property says that a ring map from R inverting every element of S factors uniquely through S^{-1}R. This explains why different denominator descriptions can yield the same localization. It also shows why elements of S become units rather than become equal to one. When R is a domain and S excludes zero, the map embeds R in its fraction field. Without the domain assumption, injectivity needs the separate condition that no nonzero element is annihilated by a member of S.

Prime ideals of S^{-1}R correspond to primes of R disjoint from S, by extension and contraction. For a prime p choose $S=R\setminus\mathfrak p$. The resulting R_p has unique maximal ideal pR_p, so it is a local ring. Its residue field is the fraction field of R/p, not generally R/p itself. For Z localized at (3), fractions have denominators not divisible by three, and the residue field is F₃. At the zero prime of a domain, localization gives the entire fraction field instead of a small neighborhood with a nonzero maximal ideal.

A local ring’s elements outside its unique maximal ideal are units. This follows because a nonunit lies in some maximal ideal, which must be the unique one. In R_p this unit criterion makes statements such as a matrix being invertible depend on whether its determinant lies outside p. Localization can therefore turn a globally noninvertible map into an invertible one. That is a feature of the microscope, not proof that the original map was an isomorphism. To detect a finite or arbitrary module’s vanishing globally, test all prime localizations, or all maximal localizations, rather than just one.

Localization is exact on modules. A concrete kernel argument clears denominators: if a localized element maps to zero, some denominator multiplier sends its numerator into the original kernel, providing a localized kernel representative. Surjectivity survives by lifting a numerator. This works without a Noetherian hypothesis. It is one of the basic examples of flatness, introduced later. Exactness means relations remain valid after this operation; it does not mean every nonzero module remains visible. A torsion module can be annihilated when its annihilating scalar is inverted.

For example, Q⊗_Z Z/6Z is zero, even though Z/6Z is nonzero. Conversely, if M is nonzero, choose a nonzero element m and a maximal ideal containing its proper annihilator. The element m remains nonzero after localization there, since no outside scalar can annihilate it. This proves that all local microscopes together detect module vanishing. The distinction between exactness and detection will return as flatness versus faithful flatness, and it prevents local computations from being promoted to global conclusions without checking which primes were discarded.