Local ring
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In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds.
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Definition and first consequences
A ring R is a local ring if it has one (and therefore all) of the following equivalent properties:
- R has a unique maximal left ideal.
- R has a unique maximal right ideal.
- 1≠0 and the sum of any two non-units in a R is a non-unit.
- 1≠0 and if x is any element of R, then x or 1−x is a unit.
- If a finite sum is a unit, then so are some of its terms (in particular the empty sum is not a unit, hence 1≠0).
If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring's Jacobson radical. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal, necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper (principal) (left) ideals I1, I2 where two ideals are called coprime if R = I1 + I2.
In the case of commutative rings, one does not have to distinguish between left, right and two-sided ideals: a commutative ring is local if and only if it has a unique maximal ideal.
Some authors require that a local ring be (left and right) Noetherian, and the non-Noetherian rings are then called "quasi-local". In this encyclopedia this requirement is not imposed.
Examples
Commutative
All fields (and skew fields) are local rings, since {0} is the only maximal ideal in these rings.
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 local behavior of these functions near 0 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 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 identify its invertible elements. A germ f is invertible if and only if f(0) ≠ 0. The reason: if f(0) ≠ 0, then 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.
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.
A more arithmetical example is the following: the ring of rational numbers with odd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. More generally, given any commutative ring R and any prime ideal P of R, the localization of R at P is local; the maximal ideal is the ideal generated by P in this localization.
Every ring of formal power series over a field (even in several variables) is local; the maximal ideal consists of those power series without constant term.
The algebra of dual numbers over any field is local. More generally, if F is a field and n is a positive integer, then the quotient ring F[X]/(Xn) is local with maximal ideal consisting of the classes of polynomials with zero constant term, since one can use a geometric series to invert all other polynomials modulo Xn. In these cases elements are either nilpotent or invertible.
Local rings play a major role in valuation theory. Given a field K, we may look for local rings in it on the assumption that it is a function field. By definition, a valuation ring of 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. 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 simplistic definition. It is replaced by using valuations.
Non-commutative
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.
Some facts and definitions
Commutative
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) 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. These are precisely the ring homomorphisms which are continuous with respect to the given topologies on R and S.
As for any topological ring, one can ask whether (R, m) is complete; if it is not, one considers its completion, again a local ring.
If (R, m) is a commutative Noetherian local ring, then
- <math>\bigcap_{i=1}^\infty m^i = \{0\}<math>
(Krull's intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space.
General
The Jacobson radical m of a local ring R (which is equal to the unique left maximal ideal and also to the unique right maximal ideal) consists precisely of the non-units of the ring; furthermore, it is the unique two-sided maximal ideal of R. (However, in the non-commutative case, having a unique two-sided maximal is not equivalent to being local.)
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.