Inmathematics, more specifically inring theory,local rings are certainrings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined onalgebraic varieties ormanifolds, or ofalgebraic number fields examined at a particularplace, or prime.Local algebra is the branch ofcommutative algebra that studiescommutative local rings and theirmodules.

In practice, a commutative local ring often arises as the result of thelocalization of a ring at aprime ideal.

The concept of local rings was introduced byWolfgang Krull in 1938 under the nameStellenringe.^[1] The English termlocal ring is due toZariski.^[2]

AringR is alocal ring if it has any one of the following equivalent properties:

• R has a uniquemaximal leftideal.
• R has a unique maximal right ideal.
• 1 ≠ 0 and the sum of any two non-units inR is a non-unit.
• 1 ≠ 0 and ifx is any element ofR, thenx or1 −x is a unit.
• If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0).

If these properties hold, then the unique maximal left ideal coincides with the unique maximal right ideal and with the ring'sJacobson radical. The third of the properties listed above says that the set of non-units in a local ring forms a (proper) ideal,^[3] necessarily contained in the Jacobson radical. The fourth property can be paraphrased as follows: a ringR is local if and only if there do not exist twocoprime proper (principal) (left) ideals, where two idealsI₁,I₂ are calledcoprime ifR =I₁ +I₂.

In the case ofcommutative 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.Before about 1960 many authors required that a local ring be (left and right)Noetherian, and (possibly non-Noetherian) local rings were calledquasi-local rings. In this article this requirement is not imposed.

A local ring that is anintegral domain is called alocal domain.

• Allfields (andskew fields) are local rings, since {0} is the only maximal ideal in these rings.
• The ring${\displaystyle \mathbb{Z}/p^n\mathbb{Z}}$ is a local ring (p prime,n ≥ 1). The unique maximal ideal consists of all multiples ofp.
• More generally, a nonzero ring in which every element is either a unit ornilpotent is a local ring.
• An important class of local rings arediscrete valuation rings, which are localprincipal ideal domains that are not fields.
• The ring${\displaystyle \mathbb{C}[[x]]}$, whose elements are infinite series$\sum _{i=0}^{\mathrm{\infty }}{a}_i{x}^i$ where multiplications are given by$(\sum _{i=0}^{\mathrm{\infty }}{a}_i{x}^i)(\sum _{i=0}^{\mathrm{\infty }}{b}_i{x}^i)=\sum _{i=0}^{\mathrm{\infty }}{c}_i{x}^i$ such that$c_n=\sum _{i+j=n}{a}_i{b}_j$, is local. Its unique maximal ideal consists of all elements that are not invertible. In other words, it consists of all elements with constant term zero.
• More generally, every ring offormal power series over a local ring is local; the maximal ideal consists of those power series withconstant term in the maximal ideal of the base ring.
• Similarly, thealgebra ofdual numbers over any field is local. More generally, ifF is a local ring andn is a positive integer, then thequotient ringF[X]/(Xⁿ) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal ofF, since one can use ageometric series to invert all other polynomialsmoduloXⁿ. IfF is a field, then elements ofF[X]/(Xⁿ) are eithernilpotent orinvertible. (The dual numbers overF correspond to the casen = 2.)
• Nonzero quotient rings of local rings are local.
• The ring ofrational numbers withodd denominator is local; its maximal ideal consists of the fractions with even numerator and odd denominator. It is the integerslocalized at 2.
• More generally, given anycommutative ringR and anyprime idealP ofR, thelocalization ofR atP is local; the maximal ideal is the ideal generated byP in this localization; that is, the maximal ideal consists of all elementsa/s witha ∈P ands ∈R -P.

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• Thering of polynomials${\displaystyle K[x]}$ over a field${\displaystyle K}$ is not local, since${\displaystyle x}$ and${\displaystyle 1-x}$ are non-units, but their sum is a unit.
• The ring of integers${\displaystyle \mathbb{Z}}$ is not local since it has a maximal ideal${\displaystyle (p)}$ for every prime${\displaystyle p}$.
• ${\displaystyle \mathbb{Z}}$/(pq)${\displaystyle \mathbb{Z}}$, wherep andq are distinct prime numbers. Both (p) and (q) are maximal ideals here.

To motivate the name "local" for these rings, we consider real-valuedcontinuous functions defined on someopen interval around${\displaystyle 0}$ of thereal line. We are only interested in the behavior of these functions near${\displaystyle 0}$ (their "local behavior") and we will therefore identify two functions if they agree on some (possibly very small) open interval around${\displaystyle 0}$. This identification defines anequivalence relation, and theequivalence classes are what are called the "germs of real-valued continuous functions at${\displaystyle 0}$". 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${\displaystyle f}$ is invertible if and only if${\displaystyle f(0)\ne 0}$. The reason: if${\displaystyle f(0)\ne 0}$, then by continuity there is an open interval around${\displaystyle 0}$ where${\displaystyle f}$ is non-zero, and we can form the function${\displaystyle g(x)=\frac{1}{f(x)}}$ on this interval. The function${\displaystyle g}$ gives rise to a germ, and the product of${\displaystyle fg}$ is equal to${\displaystyle 1}$. (Conversely, if${\displaystyle f}$ is invertible, then there is some${\displaystyle g}$ such that${\displaystyle f(0)g(0)=1}$, hence${\displaystyle f(0)\ne 0}$.)

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${\displaystyle f}$ with${\displaystyle f(0)=0}$.

Exactly the same arguments work for the ring of germs of continuous real-valued functions on anytopological space at a given point, or the ring of germs ofdifferentiable functions on anydifferentiable manifold at a given point, or the ring of germs ofrational functions on anyalgebraic variety at a given point. All these rings are therefore local. These examples help to explain whyschemes, the generalizations of varieties, are defined as speciallocally ringed spaces.

Local rings play a major role in valuation theory. By definition, avaluation ring of a fieldK is a subringR such that for every non-zero elementx ofK, at least one ofx andx⁻¹ is inR. Any such subring will be a local ring. For example, the ring ofrational numbers withodd denominator (mentioned above) is a valuation ring in${\displaystyle \mathbb{Q}}$.

Given a fieldK, which may or may not be afunction field, we may look for local rings in it. IfK were indeed the function field of analgebraic varietyV, then for each pointP ofV we could try to define a valuation ringR of functions "defined at"P. In cases whereV has dimension 2 or more there is a difficulty that is seen this way: ifF andG are rational functions onV with

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

the function

F/G

is anindeterminate form atP. Considering a simple example, such as

Y/X,

approached along a line

Y =tX,

one sees that thevalue atP is a concept without a simple definition. It is replaced by using valuations.

Non-commutative local rings arise naturally asendomorphism rings in the study ofdirect sum decompositions ofmodules over some other rings. Specifically, if the endomorphism ring of the moduleM is local, thenM isindecomposable; conversely, if the moduleM has finitelength and is indecomposable, then its endomorphism ring is local.

Ifk is afield ofcharacteristicp > 0 andG is a finitep-group, then thegroup algebrakG is local.

We also write(R,m) for a commutative local ringR with maximal idealm. Every such ring becomes atopological ring in a natural way if one takes the powers ofm as aneighborhood base of 0. This is them-adic topology onR. If(R,m) is a commutativeNoetherian local ring, then

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

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

As for any topological ring, one can ask whether(R,m) iscomplete (as auniform space); if it is not, one considers itscompletion, again a local ring. Complete Noetherian local rings are classified by theCohen structure theorem.

Inalgebraic geometry, especially whenR is the local ring of a scheme at some pointP,R /m is called theresidue field of the local ring or residue field of the pointP.

If(R,m) and(S,n) are local rings, then alocal ring homomorphism fromR toS is aring homomorphismf :R →S with the propertyf(m) ⊆n.^[4] These are precisely the ring homomorphisms that are continuous with respect to the given topologies onR andS. 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${\displaystyle \mathbb{C}[x]/(x^3)\to \mathbb{C}[x]/(x^2)}$.

TheJacobson radicalm of a local ringR (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 ofR. However, in the non-commutative case, having a unique maximal two-sided ideal is not equivalent to being local.^[5]

For an elementx of the local ringR, the following are equivalent:

• x has a left inverse
• x has a right inverse
• x is invertible
• x is not inm.

If(R,m) is local, then thefactor ringR/m is askew field. IfJ ≠R is any two-sided ideal inR, then the factor ringR/J is again local, with maximal idealm/J.

Adeep theorem byIrving Kaplansky says that anyprojective module over a local ring isfree, though the case where the module is finitely-generated is a simple corollary toNakayama's lemma. This has an interesting consequence in terms ofMorita equivalence. Namely, ifP is afinitely generated projectiveR module, thenP is isomorphic to the free moduleRⁿ, and hence the ring of endomorphisms${\displaystyle {\mathrm{E}\mathrm{n}\mathrm{d}}_R(P)}$ is isomorphic to the full ring of matrices${\displaystyle {\mathrm{M}}_n(R)}$. Since every ring Morita equivalent to the local ringR is of the form${\displaystyle {\mathrm{E}\mathrm{n}\mathrm{d}}_R(P)}$ for such aP, the conclusion is that the only rings Morita equivalent to a local ringR are (isomorphic to) the matrix rings overR.

• Lam, T.Y. (2001).A first course in noncommutative rings. Graduate Texts in Mathematics (2nd ed.). Springer-Verlag.ISBN 0-387-95183-0.
• Jacobson, Nathan (2009).Basic algebra. Vol. 2 (2nd ed.). Dover.ISBN 978-0-486-47187-7.

• Discrete valuation ring
• Semi-local ring
• Gorenstein local ring
• Regular local ring

• The philosophy behind local rings

• Ring theory
• Localization (mathematics)

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