Theorem (Elements of S become units). For every s∈S, the element ι(s)=s/1 is a unit in S−1R, with inverse 1/s. In particular, every fraction can be rewritten as
sr=ι(r)ι(s)−1.
Universal property (often used as the definition). If T is any commutative ring and φ:R→T is a homomorphism such that every φ(s) (for s∈S) is a unit of T, then there exists a unique homomorphism ψ:S−1R→T with ψ∘ι=φ. Concretely, ψ is forced to satisfy ψ(r/s)=φ(r)φ(s)−1.
This perspective explains why localizing at a prime produces a local ring: inverting all elements outside a prime ideal forces exactly those elements to become units.
Examples
Inverting a single integer. Take R=Z and S={1,2,22,23,…}. Then S−1R≅Z[1/2], and 2 becomes a unit with inverse 1/2.
Laurent polynomials by inverting a variable. Take R=k[x] and S={1,x,x2,…}. Then S−1R≅k[x,x−1], and x becomes a unit. Every element looks like a Laurent polynomial because denominators are powers of x.
Localizing at a prime ideal. If p is a prime ideal of R, set S=R∖p. Then the localization S−1R is the localization Rp, where every element not in p becomes invertible; this is the basic way to construct a local ring from R.
A commutative ring is a ringR such that ab=ba for all a,b∈R.
Let R be a commutative ring. A subset S⊆R is a multiplicative set if
1∈S, and
whenever s,t∈S, then st∈S.
Often one also assumes 0∈/S when the goal is to form the localization of a ringS−1R; if 0∈S, then 0 becomes invertible in S−1R, forcing 1=0 and hence S−1R is the zero ring.
A key source of multiplicative sets is complements of primes: if p⊂R is prime, then R∖p is multiplicative, and this choice produces the localization at a prime.
Examples
Powers of an element. For f∈R, the set
S={1,f,f2,f3,…}
is multiplicative. (If f is nilpotent, then 0∈S and the corresponding localization collapses to the zero ring.)
Complement of a prime ideal. If p is a prime ideal of R, then
S=R∖p
is multiplicative (primality ensures st∈/p whenever s,t∈/p). Localizing at this S gives Rp.
Inverting a prime number in Z. In R=Z, the subset S={1,p,p2,…} (for a prime p) is multiplicative. The localization S−1Z is the subring of Q consisting of fractions whose denominator is a power of p.
As a set, S−1R can be constructed from pairs (r,s)∈R×S modulo the equivalence relation
(r,s)∼(r′,s′)⟺∃t∈S such that t(rs′−r′s)=0 in R.
Write the class of (r,s) as sr. Addition and multiplication are defined by
sr+s′r′=ss′rs′+r′s,sr⋅s′r′=ss′rr′.
The canonical map is ι(r)=1r.
If 0∈S, then ι(0) is invertible, hence 1=0 in S−1R; in this case S−1R is the zero ring.
Universal property
The localization is characterized by the following universal mapping property:
If A is any commutative ring and φ:R→A is a ring homomorphism such that φ(s) is a unit of A for every s∈S, then there exists a unique ring homomorphism φ:S−1R→A with φ∘ι=φ. Explicitly,
A commutative ringR is a local ring if it has a unique maximal ideal. One often records this ideal and writes (R,m), where m is the unique maximal ideal.
For a commutative ring R, the following are equivalent:
R is local (i.e. it has a unique maximal ideal).
The set of nonunits in R is an ideal; this ideal is then the unique maximal ideal.
Whenever a+b=1 in R, at least one of a or b is a unit.
Local rings arise systematically from localization: if p is a prime ideal of R, then localizing at the prime produces the local ring Rp.
Many foundational results in commutative algebra are naturally stated for local rings; for instance, Nakayama's lemma is formulated for finitely generated modules over a local ring.
Examples
Fields. Any fieldk is local: its only maximal ideal is (0).
Localizing Z at a prime. For a prime number p, the ring Z(p) from localization at (p) is local, with maximal ideal pZ(p).
Localizing a polynomial ring at a maximal ideal. If k is a field, then k[x](x) is local with maximal ideal generated by x. More generally, k[x,y](x,y) is local with maximal ideal (x,y).