Local ring

To motivate the name "local" for these rings, we consider real-valued continuous functions defined on some open interval around 0 of the real line. We are only interested in the behavior of these functions near 0 (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around 0. This identification defines an equivalence relation, and the equivalence classes are what are called the "germs of real-valued continuous functions at 0". These germs can be added and multiplied and form a commutative ring.

To see that this ring of germs is local, we need to characterize its invertible elements. A germ f is invertible if and only if f(0) ≠ 0. The reason: if f(0) ≠ 0, then by continuity there is an open interval around 0 where f is non-zero, and we can form the function g(x) = 1/f(x) on this interval. The function g gives rise to a germ, and the product of fg is equal to 1. (Conversely, if f is invertible, then there is some g such that f(0)g(0) = 1, hence f(0) ≠ 0.)

With this characterization, it is clear that the sum of any two non-invertible germs is again non-invertible, and we have a commutative local ring. The maximal ideal of this ring consists precisely of those germs f with f(0) = 0.

Exactly the same arguments work for the ring of germs of continuous real-valued functions on any topological space at a given point, or the ring of germs of differentiable functions on any differentiable manifold at a given point, or the ring of germs of rational functions on any algebraic variety at a given point. All these rings are therefore local. These examples help to explain why schemes, the generalizations of varieties, are defined as special locally ringed spaces.

Local rings play a major role in valuation theory. By definition, a valuation ring of a field K is a subring R such that for every non-zero element x of K, at least one of x and x⁻¹ is in R. Any such subring will be a local ring. For example, the ring of rational numbers with odd denominator (mentioned above) is a valuation ring in ${\displaystyle \mathbb {Q} }$.

Given a field K, which may or may not be a function field, we may look for local rings in it. If K were indeed the function field of an algebraic variety V, then for each point P of V we could try to define a valuation ring R of functions "defined at" P. In cases where V has dimension 2 or more there is a difficulty that is seen this way: if F and G are rational functions on V with

F(P) = G(P) = 0,

the function

F/G

is an indeterminate form at P. Considering a simple example, such as

Y/X,

approached along a line

Y = tX,

one sees that the value at P is a concept without a simple definition. It is replaced by using valuations.

Non-commutative local rings arise naturally as endomorphism rings in the study of direct sum decompositions of modules over some other rings. Specifically, if the endomorphism ring of the module M is local, then M is indecomposable; conversely, if the module M has finite length and is indecomposable, then its endomorphism ring is local.

If k is a field of characteristic p > 0 and G is a finite p-group, then the group algebra kG is local.

We also write (R, m) for a commutative local ring R with maximal ideal m. Every such ring becomes a topological ring in a natural way if one takes the powers of m as a neighborhood base of 0. This is the m-adic topology on R. If (R, m) is a commutative Noetherian local ring, then

${\displaystyle \bigcap _{i=1}^{\infty }m^{i}=\{0\}}$

(Krull's intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space. The theorem is a consequence of the Artin–Rees lemma together with Nakayama's lemma, and, as such, the "Noetherian" assumption is crucial. Indeed, let R be the ring of germs of infinitely differentiable functions at 0 in the real line and m be the maximal ideal ${\displaystyle (x)}$. Then a nonzero function ${\displaystyle e^{-{1 \over x^{2}}}}$ belongs to ${\displaystyle m^{n}}$ for any n, since that function divided by ${\displaystyle x^{n}}$ is still smooth.

As for any topological ring, one can ask whether (R, m) is complete (as a uniform space); if it is not, one considers its completion, again a local ring. Complete Noetherian local rings are classified by the Cohen structure theorem.

In algebraic geometry, especially when R is the local ring of a scheme at some point P, R / m is called the residue field of the local ring or residue field of the point P.

If (R, m) and (S, n) are local rings, then a local ring homomorphism from R to S is a ring homomorphism f : R → S with the property f(m) ⊆ n.[4] These are precisely the ring homomorphisms which are continuous with respect to the given topologies on R and S. For example, consider the ring morphism ${\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x,y]/(x^{3},x^{2}y,y^{4})}$ sending ${\displaystyle x\mapsto x}$. The preimage of ${\displaystyle (x,y)}$ is ${\displaystyle (x)}$. Another example of a local ring morphism is given by ${\displaystyle \mathbb {C} [x]/(x^{3})\to \mathbb {C} [x]/(x^{2})}$.

The Jacobson radical m of a local ring R (which is equal to the unique maximal left ideal and also to the unique maximal right ideal) consists precisely of the non-units of the ring; furthermore, it is the unique maximal two-sided ideal of R. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.[5]

For an element x of the local ring R, the following are equivalent:

• x has a left inverse
• x has a right inverse
• x is invertible
• x is not in m.

If (R, m) is local, then the factor ring R/m is a skew field. If J ≠ R is any two-sided ideal in R, then the factor ring R/J is again local, with maximal ideal m/J.

A deep theorem by Irving Kaplansky says that any projective module over a local ring is free, though the case where the module is finitely-generated is a simple corollary to Nakayama's lemma. This has an interesting consequence in terms of Morita equivalence. Namely, if P is a finitely generated projective R module, then P is isomorphic to the free module Rⁿ, and hence the ring of endomorphisms ${\displaystyle \mathrm {End} _{R}(P)}$ is isomorphic to the full ring of matrices ${\displaystyle \mathrm {M} _{n}(R)}$. Since every ring Morita equivalent to the local ring R is of the form ${\displaystyle \mathrm {End} _{R}(P)}$ for such a P, the conclusion is that the only rings Morita equivalent to a local ring R are (isomorphic to) the matrix rings over R.