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,
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.
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.
Let R be a commutative ring, let S⊆R be a multiplicative set, and let M be an R-module. The localization of M at S is an S−1R-module, denoted S−1M, constructed so that every s∈S acts invertibly on S−1M.
Define S−1M as equivalence classes of pairs (m,s)∈M×S under
(m,s)∼(m′,s′)⟺∃t∈S such that t(s′m−sm′)=0 in M.
Write the class of (m,s) as sm. Addition is
sm+s′m′=ss′s′m+sm′,
and the scalar action of S−1R is given by
(sr)(tm)=strm.
The map ιM:M→S−1M given by ιM(m)=1m is R-linear.
Universal property
Let N be an S−1R-module. Viewing N as an R-module via the canonical map R→S−1R, every s∈S acts by an automorphism on N. The localization S−1M is characterized by:
For every R-linear map f:M→N, there exists a unique S−1R-linear map f:S−1M→N with f∘ιM=f.
In particular, localizing at a prime p means taking S=R∖p and writing
Localization interacts well with exact sequences: it is an exact functor on modules (see exactness of localization and compare with the general notion of an exact sequence).
Finally, localization can be expressed as a base change: via extension of scalars there is a natural isomorphism
S−1M≅(S−1R)⊗RM.
Examples
Localizing a quotient. If I⊆R is an ideal, then
S−1(R/I)≅(S−1R)/(S−1I),
where S−1I denotes the image of I in S−1R.
Torsion killed by localization. Take R=Z, M=Z/nZ, and localize at S=Z∖(p) (so S−1Z=Z(p)).
If p∤n, then n∈S becomes a unit, so S−1M=0.
If p∣n, then S−1M≅Z(p)/nZ(p), which is generally nonzero.
Making an element invertible forces a module to vanish. Let R=k[x], M=R/(x), and S={1,x,x2,…}. In S−1R the element x is a unit, but x annihilates M, so S−1M=0.
Let f:R→S be a homomorphism of commutative rings, and let M be an R-module. The extension of scalars (or base change) of M along f is the S-module
S⊗RM,
where S acts on the left tensor factor: s⋅(s′⊗m)=(ss′)⊗m.
There is a canonical R-linear map
ηM:M⟶S⊗RM,m⟼1⊗m,
where S⊗RM is viewed as an R-module via f.
Universal property and adjunction
For every S-module N, restriction of scalars along f produces an R-module; this is restriction of scalars. Extension of scalars is left adjoint to restriction of scalars, meaning there is a natural bijection
HomS(S⊗RM,N)≅HomR(M,ResfN),
where ResfN denotes N viewed as an R-module via f.
Quotient base change. Let S=R/I and f:R→R/I be the quotient map. Then for any R-module M,
(R/I)⊗RM≅M/IM.
For example, with R=Z, S=Z/nZ, one gets (Z/n)⊗ZM≅M/nM.
Field extension. If k⊆K is a field extension and V is a k-vector space, then K⊗kV is the K-vector space obtained by extending scalars. If V≅kd is finite-dimensional, then K⊗kV≅Kd.
Localization as extension of scalars. Let R=k[x], let S={1,x,x2,…}, and set R′=S−1R≅k[x,x−1]. For M=R/(x), extension of scalars gives
R′⊗RM≅S−1M=0,
since x becomes invertible after localization but kills M.
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).