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Local ring

In commutative algebra, a local ring is a commutative ring R that admits exactly one maximal ideal \mathfrak{m}_R, which consists precisely of all non-unit elements of R. The residue field of such a ring is the quotient R / \mathfrak{m}_R, often denoted \kappa(R), and elements of R outside \mathfrak{m}_R are invertible. This structure captures the "local" behavior of more general rings, where the non-units form an ideal, distinguishing local rings from rings with multiple maximal ideals. Local rings arise naturally through the process of localization: for any A and \mathfrak{p} \subset A, the localization A_\mathfrak{p} is a local ring with \mathfrak{p} A_\mathfrak{p} and residue field the fraction field of A / \mathfrak{p}. Fields are trivial examples of local rings (with \mathfrak{m}_R = \{0\}), and more generally, the spectrum \operatorname{Spec}(R) of a local ring consists of a unique closed point corresponding to \mathfrak{m}_R. Equivalent characterizations include the condition that for every element x \in R, either x or $1 - x is invertible (assuming R is not the zero ring), highlighting the dichotomy between units and non-units. Local rings are foundational in , serving as the "bread and butter" for studying schemes and varieties by focusing on behavior at individual points. They enable key tools like completions (yielding complete local rings for in p-adic analysis) and homological methods, such as , which governs the structure of finitely generated modules over local rings. Important subclasses include discrete valuation rings (regular local rings of dimension 1), regular local rings (where the maximal ideal is generated by a of length equal to the ), and Gorenstein rings (with finite injective dimension). These structures underpin local cohomology, dimension theory, and resolutions in both and .

Definition and Basic Properties

Definition

In , a is a with unity that possesses exactly one , denoted \mathfrak{m}. The elements of \mathfrak{m} are precisely the non-units of the ring, and the R / \mathfrak{m} forms a , known as the of R, often denoted k. This definition assumes familiarity with basic concepts in , such as rings equipped with a multiplicative , ideals, and maximal ideals, where a maximal ideal is a proper ideal not contained in any larger proper ideal. In the non-commutative setting, a R with unity is local if its Jacobson radical J(R), the intersection of all maximal left ideals, coincides with the set of non-units and R / J(R) is a . Equivalently, R has a unique maximal left ideal (or, symmetrically, a unique maximal right ideal), which serves as the Jacobson radical.

Characterization and Consequences

In a ring R with identity, the non-units form an ideal if and only if R is local, meaning it possesses a unique maximal (two-sided) ideal \mathfrak{m}, which coincides precisely with the set of all non-units. Equivalently, R is local if the sum of any two non-units is itself a non-unit, a condition that ensures every element of R is either a unit or belongs to \mathfrak{m}. This characterization extends to one-sided ideals: R is local if and only if it has a unique maximal right ideal or a unique maximal left ideal. A direct consequence of this structure is that if a + b is a unit in R, then at least one of a or b must be a unit; otherwise, both would lie in \mathfrak{m}, implying their sum also belongs to \mathfrak{m} and thus cannot be a unit. Furthermore, the unique maximal ideal \mathfrak{m} is the Jacobson radical J(R) of R, as it is the of all maximal ideals (of which there is only one). Local rings also admit no non-trivial idempotents: the only idempotent elements are $0 and $1. In the commutative case, the set of non-invertible elements is exactly the unique \mathfrak{m}, reinforcing that units are precisely the elements outside \mathfrak{m}. The R/\mathfrak{m} then forms a , known as the of R. In the commutative case, to see that \mathfrak{m} comprises all non-units, suppose x \in R is a non-unit not in \mathfrak{m}; then the ideal generated by x and \mathfrak{m} is proper (as x is non-invertible), hence contained in some , but by uniqueness this must be \mathfrak{m}, implying x \in \mathfrak{m}, a . For the unit property, assume neither a nor b is a unit, so both are in \mathfrak{m}; their lies in \mathfrak{m} by ideal closure, hence is non-invertible. The absence of non-trivial idempotents follows from the fact that a non-trivial idempotent e would allow a decomposition R = Re \oplus R(1-e) as modules over R, contradicting the local property (indecomposability). Fields provide the trivial local ring example, where \mathfrak{m} = (0) is the unique maximal ideal. The zero ring, however, is excluded from consideration as local, since it lacks a unique maximal ideal (or violates the identity requirement in standard definitions).

Examples

Commutative Examples

One fundamental example of a commutative local ring is the ring of k[] over a k. This ring consists of all infinite series \sum_{i=0}^\infty a_i x^i with coefficients a_i \in k, equipped with the usual addition and multiplication of series. It is a local ring with unique (x), generated by x, and the is k[]/(x) \cong k. Another class of examples arises from quotients of rings. For a k and n \geq 1, the k/(x^n) is commutative and local, with unique (x)/(x^n), and k/(x^n)/((x)/(x^n)) \cong k. Elements outside this are units, confirming the local structure. The of p-adic integers \mathbb{Z}_p, for a prime p, provides a key example from . Defined as the completion of \mathbb{Z} with respect to the p-adic valuation, \mathbb{Z}_p is a commutative local with unique p\mathbb{Z}_p, and \mathbb{Z}_p / p\mathbb{Z}_p \cong \mathbb{F}_p. In , local rings appear as coordinate rings at points on varieties. Consider the affine plane over a k, with coordinate ring k[x,y]; the localization at the (x,y) yields the local ring k[x,y]_{(x,y)}, which is and local with unique (x,y)k[x,y]_{(x,y)}, and isomorphic to k. This ring captures the local structure at the origin (0,0). A general construction of commutative local rings uses localization at s. For any R and \mathfrak{p} \subset R, the localization R_\mathfrak{p} is a commutative local ring with unique \mathfrak{p} R_\mathfrak{p}, and R_\mathfrak{p} / \mathfrak{p} R_\mathfrak{p} \cong \mathrm{Frac}(R/\mathfrak{p}). This process inverts all elements outside \mathfrak{p}, ensuring the local property.

Non-Commutative Examples

In non-commutative algebra, division rings provide the simplest examples of local rings, as they possess no proper nonzero left ideals, making the zero ideal the unique maximal left ideal (with Jacobson zero). For instance, the quaternions over the reals form a non-commutative that is local in this sense. Full matrix rings M_n(D) over a D with n > 1 are not local, as they admit multiple maximal left ideals; their Jacobson is zero (being Artinian rings), but the semisimple M_n(D) has n^2 pairwise non-isomorphic left modules, corresponding to distinct maximal left ideals. In contrast, when n=1, M_1(D) \cong D recovers the local division ring case. Group rings offer another class of non-commutative local rings: if k is a field of characteristic p > 0 and G is a finite p-group, then the group ring k[G] is local, with the augmentation ideal \Delta(k[G]) = \{ \sum_{g \in G} a_g g \mid \sum_{g \in G} a_g = 0 \} serving as the unique maximal left ideal (and Jacobson radical). This ideal is nilpotent of index |G|, and the quotient k[G]/\Delta(k[G]) \cong k is a division ring. Artinian local rings provide finite-dimensional examples beyond group rings. Consider the ring T_n(D) of n \times n upper triangular matrices over a D with constant diagonal entries (i.e., all diagonal elements equal). This ring is non-commutative for n > 1 and local, with Jacobson radical consisting of the strictly upper triangular matrices ( of index n), and the quotient T_n(D)/J(T_n(D)) \cong D a , ensuring a unique maximal left . The Weyl algebra A_1(k) = k\langle x, \partial \rangle over a field k (with relation \partial x - x \partial = 1) is a non-commutative example where the Jacobson radical is zero (as it is simple), but it is not local in the strict sense, possessing infinitely many distinct maximal left ideals despite the unique maximal two-sided ideal being zero.

Non-Examples

Rings that fail to be local provide insight into the structural conditions required for locality, primarily by exhibiting either no maximal ideals or more than one. A fundamental reason for non-locality is the presence of multiple maximal ideals, which often arises from the ring's ability to "decompose" into components corresponding to distinct "points" or prime elements. This contrasts with local rings, where all non-units are contained in a single maximal ideal, enabling focused study of local behavior such as completions or valuations. Below, several canonical non-examples are discussed, drawn from standard commutative algebra. Consider the polynomial ring k over a k. If k is infinite, k possesses infinitely many distinct maximal ideals of the form (x - a) for each a \in k; even for finite k, there are |k| such ideals, exceeding one unless |k| = 1. These maximal ideals correspond to at distinct points, reflecting the affine line's multiple points in , which makes k suitable for global properties like unique factorization but precludes locality. Direct products of rings illustrate another common failure mode. For nonzero rings R and S each with at least one , the product ring R \times S has at least two , including \mathfrak{m} \times S for any \mathfrak{m} of R and R \times \mathfrak{n} for any \mathfrak{n} of S. This multiplicity stems from the 's decomposition into independent components, useful for studying direct sums or disjoint unions but incompatible with the unified non-unit structure of local rings. The , where the additive and multiplicative identities coincide (i.e., $0 = 1), admits no proper s whatsoever and thus has no maximal ideals, failing the condition for being . Similarly, certain rings with zero divisors, such as \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}, exhibit exactly two maximal ideals: the principal ideals generated by (1,0) and (0,1), respectively. These examples highlight rings where the absence or proliferation of maximal ideals disrupts the unique "local" focus, yet they remain valuable for modeling trivial or discrete structures without a dominant non-unit . Integral domains like the \mathbb{Z} also serve as non-local examples, featuring infinitely many maximal ideals (p) for each p. This infinitude arises from the abundance of prime elements, allowing \mathbb{Z} to capture arithmetic globally across all primes, in contrast to local domains that zoom in on a single prime. In general, non-locality often signals a ring's capacity for multiple irreducible components or points, making such rings essential for broader algebraic and geometric investigations despite lacking a maximal ideal.

Commutative Local Rings

Valuation Rings

A valuation ring is an R with fraction field K equipped with a valuation v: K^\times \to \Gamma, where \Gamma is a totally ordered , such that R = \{ x \in K \mid v(x) \geq 0 \} \cup \{0\} and the is \mathfrak{m} = \{ x \in K \mid v(x) > 0 \}. The valuation v satisfies v(xy) = v(x) + v(y) and v(x + y) \geq \min(v(x), v(y)) for all x, y \in K^\times. Valuation rings are local rings with maximal ideal \mathfrak{m}, and the residue field k = R / \mathfrak{m} is a field. The valuation induces a topology on K, known as the valuation topology, where the basic open sets are defined using balls around elements based on v. Moreover, R is a normal domain, meaning it is integrally closed in K. An equivalent characterization is that R is a valuation ring if and only if for every x \in K^\times, either x \in R or x^{-1} \in R. Another equivalent condition is that R is an integral domain such that for any a, b \in R, either a divides b or b divides a in R. Examples include discrete valuation rings (DVRs), which arise from rank-one valuations where \Gamma \cong \mathbb{Z}. The ring \mathbb{Z}_{(p)} = \{ a/b \in \mathbb{Q} \mid p \nmid b \} for a prime p, with valuation v_p, is a DVR. Similarly, the formal power series ring k[] over a field k, with valuation v(t) = 1, is a DVR. In scheme theory, the spectrum \operatorname{Spec} R of a valuation ring R consists of a totally ordered chain of prime ideals corresponding to the convex subgroups of \Gamma, featuring a closed point (the maximal ideal \mathfrak{m}) and the generic point (the zero ideal).

Rings of Power Series and Germs

In commutative algebra, the ring of formal power series k[[x_1, \dots, x_n]] over a field k in n indeterminates is a fundamental example of a local ring, with unique maximal ideal \mathfrak{m} = (x_1, \dots, x_n) consisting of series with zero constant term. This ring is complete with respect to the \mathfrak{m}-adic topology, meaning every Cauchy sequence in this topology converges, which endows it with a natural topological structure suitable for studying limits of ideals and modules. Rings of germs arise in analytic contexts as local models for functions near a point. For instance, the ring of germs of holomorphic functions at the origin in \mathbb{C}^n, denoted \mathcal{O}_{\mathbb{C}^n, 0}, comprises equivalence classes of holomorphic functions defined in neighborhoods of the origin, where two functions are equivalent if they agree on some common neighborhood. This ring is local, with maximal ideal \mathfrak{m} formed by germs vanishing at the origin, i.e., functions f such that f(0) = 0. Similarly, the ring of germs of C^\infty functions at a point in \mathbb{R}^n is local, with maximal ideal consisting of smooth functions vanishing at that point, capturing infinitesimal behavior in differential geometry. The construction of these rings distinguishes formal power series from convergent ones in . k[[x_1, \dots, x_n]] allow arbitrary coefficients without convergence requirements, serving as algebraic completions, whereas the ring of convergent power series \mathbb{C}\{x_1, \dots, x_n\} consists of series with positive , forming a that embeds densely into the formal series ring. These rings exhibit strong algebraic properties: k[[x_1, \dots, x_n]] is Noetherian, meaning every ideal is finitely generated, and has n, equal to the number of indeterminates. Moreover, if k is a of characteristic zero, this ring is , ensuring good behavior under completions and localizations, such as finite integral extensions remaining Noetherian. The ring of holomorphic germs \mathcal{O}_{\mathbb{C}^n, 0} shares these traits, being Noetherian and of dimension n. Geometrically, the local ring \mathcal{O}_{X,p} at a point p on an algebraic variety X is the stalk of the structure sheaf, isomorphic to a ring of power series or germs that models the infinitesimal neighborhood of p, encoding tangent spaces and higher-order approximations via the maximal ideal powers \mathfrak{m}^k. This structure facilitates the study of singularities and deformations, where formal power series capture algebraic aspects and germs incorporate analytic ones. A related notion involves étale local rings, which refine these models through étale morphisms, providing étale neighborhoods that locally resemble power series rings while preserving exactness in cohomology.

Discrete Valuation Rings

A discrete valuation ring (DVR) is a valuation ring whose value group is isomorphic to the integers \mathbb{Z}. It is a (PID) that is , with its unique nonzero \mathfrak{m} generated by a uniformizer \pi, so \mathfrak{m} = (\pi). The associated valuation v: K^\times \to \mathbb{Z} on the fraction field K satisfies v(\pi) = 1, and every nonzero ideal of the DVR is of the form (\pi^n) for some n \geq 0. DVRs are Noetherian integrally closed domains of Krull dimension 1. As PIDs, all their ideals are principal, and their fraction fields are equipped with the discrete valuation that extends additively and respects the minimum property for sums. They are regular local rings, meaning the maximal ideal is generated by a regular sequence of length equal to the dimension. Prominent examples include the ring of formal power series k[] over a field k, where t serves as the uniformizer and the maximal ideal is (t). Another is the ring of p-adic integers \mathbb{Z}_p, the completion of \mathbb{Z}_{(p)} at the prime p, with uniformizer p and residue field \mathbb{F}_p. In algebraic number theory, the localization of the ring of integers \mathcal{O}_K of a number field K at a nonzero prime ideal \mathfrak{p} yields a DVR, with uniformizer a uniformizing element of \mathfrak{p} and residue field the finite field \mathcal{O}_K / \mathfrak{p}. In a DVR with uniformizer \pi, every nonzero element x \in K can be uniquely expressed as x = u \pi^e, where u is a in the DVR and e = v(x) \in \mathbb{Z} is the order of x. This decomposition facilitates unique factorization and enables Euclidean-like algorithms for computing greatest common divisors in the ring. A Noetherian local domain of dimension 1 is a DVR if and only if it is . Equivalently, it is (integrally closed) and has a principal . By the Cohen structure theorem, every complete Noetherian DVR of dimension 1 is isomorphic to a power series ring over a field or a Cohen ring (a complete DVR with prime uniformizer). Thus, complete DVRs are precisely the complete regular local rings of dimension 1.

General Local Rings

Nakayama's Lemma

Nakayama's lemma is a fundamental result in module theory over local rings, providing a criterion for when submodules or ideals can be lifted from the residue field back to the original module. Let (R, \mathfrak{m}) be a local ring with maximal ideal \mathfrak{m} and residue field k = R/\mathfrak{m}. For a finitely generated R-module M, the lemma states that if \mathfrak{m}M = M, then M = 0. More generally, if I \subseteq \mathfrak{m} is an such that IM = M, then M = 0. A useful follows: if M = N + IM for some submodule N \subseteq M, then M = N. These statements hold because \mathfrak{m} is contained in the Jacobson radical of R, ensuring that elements of $1 + \mathfrak{m} are s in R. The proof of the basic form proceeds in two ways. First, using the Cayley-Hamilton theorem or a argument: since M is finitely generated, say by y_1, \dots, y_n, the relation y_i = \sum_j z_{ij} y_j with z_{ij} \in I yields a A = ( \delta_{ij} - z_{ij} ) with entries in $1 + I. The f = \det(A) lies in $1 + I, hence is a unit, and applying the shows fM = 0, implying M = 0. Alternatively, assume a minimal generating set \{u_1, \dots, u_n\} for M \neq 0. Then u_n \in IM = \sum a_i u_i with a_i \in I \subseteq \mathfrak{m}, so u_n (1 - a_n) \in \sum_{i < n} a_i' u_i. Since $1 - a_n is a unit, this contradicts minimality unless n=0, hence M=0. Important corollaries include the invariance of the minimal number of generators: if \{x_1, \dots, x_d\} minimally generate M, then \{\bar{x}_1, \dots, \bar{x}_d\} form a basis for the k-vector space M/\mathfrak{m}M, so d = \dim_k (M/\mathfrak{m}M). This quantity, often denoted \mu(M), is the minimal number of generators of M. Another consequence concerns supports: for finitely generated M, the annihilator \mathrm{Ann}_R(M) contains no prime ideal disjoint from the support of M/\mathfrak{m}M. Nakayama's lemma is crucial for proving the Hilbert basis theorem, which states that if R is Noetherian, then so is R, by showing that ideals lift appropriately from residue fields. It also facilitates the study of syzygies in minimal free resolutions over rings, ensuring that relations modulo \mathfrak{m} determine the structure. The lemma extends to non-commutative rings: for a ring R with Jacobson radical J(R), and a finitely generated right R-module M, if J(R)M = M, then M = 0. This holds in particular for non-commutative rings, defined as rings with a unique maximal right ideal (which coincides with J(R)), provided J(R) is Artinian to ensure the proofs adapt via or nilpotency arguments.

Localizations and Completions

In , localization at a provides a fundamental construction for obtaining rings. Given a R and a \mathfrak{p} \subseteq R, the localization R_{\mathfrak{p}} is formed by inverting the multiplicative set S = R \setminus \mathfrak{p}, yielding the ring of fractions S^{-1}R. This ring R_{\mathfrak{p}} is , with unique maximal ideal \mathfrak{p} R_{\mathfrak{p}}, consisting of fractions a/s where a \in \mathfrak{p} and s \in S. The natural map R \to R_{\mathfrak{p}} sends elements of S to units, and R_{\mathfrak{p}} satisfies a universal property: for any ring homomorphism f: R \to B such that f(S) consists of units in B, there exists a unique extension \tilde{f}: R_{\mathfrak{p}} \to B with \tilde{f}(r/1) = f(r) for all r \in R. Moreover, if R is an integral domain, then R_{\mathfrak{p}} is flat over R. Completion offers another key method to construct or refine local rings, particularly in the Noetherian setting. For a Noetherian local ring (R, \mathfrak{m}), the \mathfrak{m}-adic completion \hat{R} is the inverse limit \hat{R} = \lim_{\leftarrow n} R / \mathfrak{m}^n, equipped with the \mathfrak{m}-adic topology. The canonical map R \to \hat{R} is flat, and if \mathfrak{m} lies in the Jacobson radical, it is faithfully flat. This completion preserves Noetherianity and exactness for finite modules, making \hat{R} a useful tool for studying properties preserved under completion, such as those verified via Nakayama's lemma in one sentence. The Cohen structure theorem provides a precise description of complete local Noetherian rings. For a complete local Noetherian ring (R, \mathfrak{m}) with residue field k = R / \mathfrak{m}, if \mathfrak{m} is finitely generated, then R is isomorphic to a quotient of a power series ring over a coefficient ring: specifically, R \cong \Lambda [[x_1, \dots, x_d]] / I, where \Lambda is either a field or a complete discrete valuation ring (Cohen ring) with residue field k, and I is an ideal. This theorem classifies such rings up to their embedding dimension d and highlights their quotient structure from regular local rings. In the non-commutative setting, analogous notions of localization and completion exist but are more complex due to the lack of commutativity. For non-commutative rings, localization requires the Ore condition on the multiplicative set to ensure well-defined fractions, and completions, such as I-adic completions for ideals I, preserve Noetherianity under additional hypotheses like polycentrality of I. Ore extensions, which adjoin indeterminates with derivations or automorphisms to a base ring, provide examples of non-commutative rings where such completions can be studied, often leading to structures with similar local properties but requiring careful handling of left and right ideals. Many prominent local rings emerge from these constructions: for instance, rings of germs arise as localizations of coordinate rings at maximal ideals corresponding to points, while p-adic integers form the completion of \mathbb{Z} at the prime (p). These processes underscore the role of localization and completion in generating local rings central to and .

Krull Dimension

In , the Krull dimension of a local ring (R, \mathfrak{m}) is defined as the supremum of the lengths of strictly ascending chains of contained in \mathfrak{m}, which coincides with the height of the \mathfrak{m}. This measure captures the "size" of the ring in terms of its prime ideal structure and is finite for Noetherian local rings. For Noetherian local rings, the Krull dimension equals the transcendence degree of the fraction field of R over the fraction field of R/\mathfrak{m}, assuming R/\mathfrak{m} is a . By the Hilbert-Krull theorem, this dimension also equals the Krull dimension of a of a over the residue R/\mathfrak{m} in d variables modulo some , where d relates to the embedding . In regular local rings, the Krull dimension equals the embedding dimension, defined as the minimal number of generators of \mathfrak{m}. A Cohen-Macaulay local ring satisfies the condition that its depth equals its , where depth is the length of the longest in \mathfrak{m}. The Auslander-Buchsbaum formula relates these invariants: for a finitely generated M over a commutative Noetherian local ring R with finite projective dimension, \mathrm{pd}_R M = \mathrm{depth} R - \mathrm{depth} M. Examples illustrate these concepts: the power series ring k[[x, y]] over a k has 2, as it admits a of primes (0) \subset (x) \subset (x, y). A has dimension 1, with prime ideals (0) and the maximal ideal. , as local rings with \mathfrak{m} = (0), have 0. For non-commutative local rings, the is less standard and often replaced by invariants like the left global dimension, which measures the supremum of projective dimensions of left modules; in cases, it may coincide with a chain length when definable.