# Local ring

A local ring is a ring that has exactly one maximal ideal. The maximal ideal is usually written \( \mathfrak{m} \), and the field \( R/\mathfrak{m} \) is called the residue field; one often writes "let \( (R, \mathfrak{m}) \) be a local ring" or "let \( (R, \mathfrak{m}, \kappa) \) be a local ring" to fix this data at the outset.<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup> Local rings are the basic algebraic device for studying a geometric object near a single point, and they pervade commutative algebra and algebraic geometry.<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup>

| Key fact | Detail |
|---|---|
| Definition | A ring with exactly one maximal ideal \( \mathfrak{m} \); the quotient \( \kappa = R/\mathfrak{m} \) is the residue field<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup> |
| Unit criterion | \( R \) is local if and only if it has a maximal ideal \( \mathfrak{m} \) and every element of \( R \setminus \mathfrak{m} \) is a unit; equivalently, for every \( x \in R \), either \( x \) or \( 1 - x \) is invertible<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup> |
| Geometric form | \( R \) is local if and only if \( \mathrm{Spec}(R) \) has exactly one closed point<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup> |
| Canonical example | For any prime ideal \( \mathfrak{p} \) of a ring \( R \), the localization \( R_{\mathfrak{p}} \) is local with maximal ideal \( \mathfrak{p}R_{\mathfrak{p}} \), and every local ring arises this way<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup><sup> • </sup><sup>[2](https://math.uchicago.edu/~amathew/chthreeimportantfunctors.pdf)</sup> |
| Fields | A field is a local ring (its maximal ideal is \( (0) \)), and any ring map between fields is a local homomorphism<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup><sup> • </sup><sup>[2](https://math.uchicago.edu/~amathew/chthreeimportantfunctors.pdf)</sup> |
| Key tool | Nakayama's lemma: if \( (A, \mathfrak{m}) \) is local and \( M \) is a finitely generated module with \( M = \mathfrak{m}M \), then \( M = 0 \)<sup>[3](https://public.websites.umich.edu/~mmustata/ag-1213-2017.pdf)</sup> |
| Noetherian convention | Some authors define a local ring to be Noetherian by fiat and call non-Noetherian ones quasilocal; the Stacks project and most modern texts do not require Noetherianity in the definition<sup>[4](https://people.math.osu.edu/snapp.14/IntroCommAlg.pdf)</sup> |

## Equivalent characterizations

Several conditions on a ring \( R \) are equivalent to locality, and each gives a different working handle on the concept.<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup>

1. \( R \) has exactly one maximal ideal.
2. \( \mathrm{Spec}(R) \) has exactly one closed point.
3. \( R \) has a maximal ideal \( \mathfrak{m} \) and every element of \( R \setminus \mathfrak{m} \) is a unit.
4. \( R \) must be nonzero, and each \( x \in R \) must have the property that at least one of \( x \) and \( 1 - x \) admits an inverse, possibly both.

The third condition explains the structure of a local ring: the nonunits form exactly the maximal ideal \( \mathfrak{m} \), so \( \mathfrak{m} \) collects everything that cannot be inverted. The fourth condition is the most computational; it follows from the [Chinese remainder theorem](https://www.edgepedia.org/chinese-remainder-theorem) that two distinct maximal ideals would produce an \( x \) with neither \( x \) nor \( 1 - x \) invertible.<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup> A related consequence is that a local ring has no idempotents other than 0 and 1, so a product of two nonzero rings is never local.<sup>[2](https://math.uchicago.edu/~amathew/chthreeimportantfunctors.pdf)</sup>

## Localization at a prime

The source of essentially all local rings is localization. If \( \mathfrak{p} \) is a prime ideal of \( R \), inverting every element outside \( \mathfrak{p} \) produces the localization \( R_{\mathfrak{p}} \), and every prime ideal of \( R_{\mathfrak{p}} \) is contained in \( \mathfrak{p}R_{\mathfrak{p}} \); hence \( \mathfrak{p}R_{\mathfrak{p}} \) is the unique maximal ideal.<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup> The residue field \( \kappa(\mathfrak{p}) = R_{\mathfrak{p}}/\mathfrak{p}R_{\mathfrak{p}} \) can be identified with the field of fractions of the domain \( R/\mathfrak{p} \).<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup><sup> • </sup><sup>[4](https://people.math.osu.edu/snapp.14/IntroCommAlg.pdf)</sup> Conversely, any local ring is of the form \( R_{\mathfrak{p}} \) for some ring \( R \) and prime \( \mathfrak{p} \), for instance by taking \( R \) to be the ring itself and \( \mathfrak{p} \) its maximal ideal.<sup>[2](https://math.uchicago.edu/~amathew/chthreeimportantfunctors.pdf)</sup>

A concrete example is \( \mathbb{Z}_{(p)} \), the localization of the integers at a prime number \( p \): it consists of the rational numbers whose denominator is not divisible by \( p \), and its maximal ideal is the set of fractions whose numerator is divisible by \( p \).<sup>[2](https://math.uchicago.edu/~amathew/chthreeimportantfunctors.pdf)</sup> The integers themselves are not local, since \( (p) \) is maximal for each prime \( p \).<sup>[2](https://math.uchicago.edu/~amathew/chthreeimportantfunctors.pdf)</sup>

## Local homomorphisms

A ring map \( \varphi : R \to S \) between local rings is a local homomorphism when \( \varphi(\mathfrak{m}_R) \subset \mathfrak{m}_S \).<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup> This condition has two useful reformulations: the preimage of \( \mathfrak{m}_S \) equals \( \mathfrak{m}_R \), and a ring element \( x \in R \) is invertible in \( R \) whenever \( \varphi(x) \) is invertible in \( S \); that is, the map reflects units.<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup> For any ring map \( \varphi : R \to S \) and any prime \( \mathfrak{q} \subset S \) lying over \( \mathfrak{p} = \varphi^{-1}(\mathfrak{q}) \), the induced map \( R_{\mathfrak{p}} \to S_{\mathfrak{q}} \) is automatically local, and it induces a map of residue fields \( \kappa(\mathfrak{p}) \to \kappa(\mathfrak{q}) \).<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup> By contrast, if \( \mathfrak{p} \) is a non-maximal prime of a local ring \( R \), the localization map \( R \to R_{\mathfrak{p}} \) is not a local homomorphism.<sup>[1](https://stacks.math.columbia.edu/tag/07BH)</sup>

## Nakayama's lemma

The single most used consequence of locality is Nakayama's lemma: if \( (A, \mathfrak{m}) \) is a local ring and \( M \) is a finitely generated module with \( M = \mathfrak{m}M \), then \( M = 0 \).<sup>[3](https://public.websites.umich.edu/~mmustata/ag-1213-2017.pdf)</sup> Equivalently, a finitely generated module \( M \) is zero as soon as its reduction \( M/\mathfrak{m}M \) over the residue field is zero, so questions about generators and surjections can be checked after reducing modulo \( \mathfrak{m} \).<sup>[2](https://math.uchicago.edu/~amathew/chthreeimportantfunctors.pdf)</sup> The Stacks project records the attribution: the lemma is due to T. Nakayama, G. Azumaya, and W. Krull, the priority is obscure, and Nakayama himself did not like the name.<sup>[5](https://stacks.math.columbia.edu/tag/07RC)</sup>

Nakayama's lemma is one reason mathematicians deliberately reduce problems to local rings. A module homomorphism \( f : M \to N \) is injective if and only if every localization \( f_{\mathfrak{m}} : M_{\mathfrak{m}} \to N_{\mathfrak{m}} \) at a maximal ideal is injective, and localization is an exact functor, so many statements about rings and modules can be proved one maximal ideal at a time.<sup>[2](https://math.uchicago.edu/~amathew/chthreeimportantfunctors.pdf)</sup>

## Local rings in algebraic geometry

In algebraic geometry, the local ring \( \mathcal{O}_{X,x} \) of a variety or scheme at a point \( x \) consists of the germs of regular functions near \( x \); concretely, for an affine variety with coordinate ring \( A(X) \), it is the ring of fractions \( f/g \) with \( g(P) \neq 0 \).<sup>[6](https://calwoo.github.io/notes/concepts/algebraic-geometry/shafarevich-ch2-local-properties/index.html)</sup> It gives a language for statements of the form "such and such holds in some open set containing \( x \)" without naming the open set.<sup>[7](https://ananddeopurkar.org/teaching/2021_algebraic_geometry/notes/)</sup> The isomorphism class of \( \mathcal{O}_{X,x} \) is an invariant of the germ \( (X, x) \), and properties such as its dimension or regularity are étale-local properties of the germ.<sup>[8](https://stacks.math.columbia.edu/tag/04QT)</sup>

**Smoothness and completion.** At a smooth point \( x \) of a variety, the completion of the local ring with respect to its maximal ideal is a formal power series ring \( k[[T_1, \ldots, T_n]] \) with \( n = \dim X \), and the local ring at a smooth point is a domain and a unique factorization domain.<sup>[9](https://math.uchicago.edu/~amathew/AGnotes.pdf)</sup> Geometrically, a smooth point lies on only one irreducible component of the variety.<sup>[9](https://math.uchicago.edu/~amathew/AGnotes.pdf)</sup> The local rings themselves remember more than the analytic topology: two points of a smooth connected complex variety have isomorphic neighborhoods as complex manifolds, but their local rings are generally not isomorphic, while their completions at smooth points are.<sup>[9](https://math.uchicago.edu/~amathew/AGnotes.pdf)</sup>

**Completion as a tool.** For a Noetherian local ring \( (R, \mathfrak{m}) \), the \( \mathfrak{m} \)-adic completion \( \varprojlim_n R/\mathfrak{m}^n \) is faithfully flat over \( R \), which lets one transfer information between the ring and its completion.<sup>[10](https://stacks.math.columbia.edu/tag/0BNH)</sup> The passage is not innocuous in every respect: the Stacks project constructs a Noetherian local domain of dimension 2, complete with respect to a principal ideal, whose recompletion after a localization is nonreduced, showing that completion can introduce nilpotents absent from the original domain.<sup>[11](https://stacks.math.columbia.edu/tag/0GHH)</sup>

## History

The theory of local rings is usually traced to [Wolfgang Krull](https://www.edgepedia.org/wolfgang-krull)'s paper "Dimensionstheorie in Stellenringen" (Journal für die reine und angewandte Mathematik 179, 1938, pp. 204–226) and to [Claude Chevalley](https://www.edgepedia.org/claude-chevalley)'s "On the theory of local rings" ([Annals of Mathematics](https://www.edgepedia.org/annals-of-mathematics) 44, 1943, pp. 690–708).<sup>[12](https://mathoverflow.net/questions/21903/origin-of-the-term-localization-for-the-localization-of-a-ring)</sup><sup> • </sup><sup>[13](https://www.jstor.org/stable/i307217)</sup> Oscar Zariski's 1946 paper "Generalized semi-local rings" attributes the theory to these two works, and Chevalley's paper, which develops localization at a multiplicative set, is regarded as a reasonable guide to the situation; Chevalley was motivated by algebraic geometry.<sup>[12](https://mathoverflow.net/questions/21903/origin-of-the-term-localization-for-the-localization-of-a-ring)</sup> The terminology "localization" itself does not appear in these early papers, and its exact origin is not settled by the documentary record.<sup>[12](https://mathoverflow.net/questions/21903/origin-of-the-term-localization-for-the-localization-of-a-ring)</sup>

## References

1. [The Stacks Project, Algebra, Section 10.18: Local rings](https://stacks.math.columbia.edu/tag/07BH)
2. [CRing Project, Chapter 3 (David Madore, Anton Geraschenko et al.)](https://math.uchicago.edu/~amathew/chthreeimportantfunctors.pdf)
3. [Mircea Mustață, Nakayama's lemma and Krull's intersection theorem, course notes, University of Michigan](https://public.websites.umich.edu/~mmustata/ag-1213-2017.pdf)
4. [Introduction to Commutative Algebra (course notes), Ohio State University](https://people.math.osu.edu/snapp.14/IntroCommAlg.pdf)
5. [The Stacks Project, Algebra, Section 10.20: Nakayama's lemma](https://stacks.math.columbia.edu/tag/07RC)
6. [Calvin Woo, Local Properties of Varieties (notes on Shafarevich, Chapter 2)](https://calwoo.github.io/notes/concepts/algebraic-geometry/shafarevich-ch2-local-properties/index.html)
7. [Anand Deopurkar, Algebraic Geometry lecture notes (2021)](https://ananddeopurkar.org/teaching/2021_algebraic_geometry/notes/)
8. [The Stacks Project, Schemes, Section 35.21: Local properties of germs](https://stacks.math.columbia.edu/tag/04QT)
9. [Matthew Emerton, Notes on algebraic geometry, Lectures 19–20, University of Chicago](https://math.uchicago.edu/~amathew/AGnotes.pdf)
10. [The Stacks Project, Algebra, Section 10.97: Completion for Noetherian rings](https://stacks.math.columbia.edu/tag/0BNH)
11. [The Stacks Project, Examples, Section 110.18: Another local ring with nonreduced completion](https://stacks.math.columbia.edu/tag/0GHH)
12. [MathOverflow: Origin of the term "localization" for the localization of a ring](https://mathoverflow.net/questions/21903/origin-of-the-term-localization-for-the-localization-of-a-ring)
13. [Annals of Mathematics, Vol. 44, No. 4 (October 1943), table of contents (JSTOR)](https://www.jstor.org/stable/i307217)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Commutative algebra › Localization and local rings*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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