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Maximal ideal

In ring theory, a maximal ideal of a commutative ring R with unity is a proper ideal M (i.e., M \neq R) such that no other proper ideal of R strictly contains M.^[1] Equivalently, M is maximal if and only if the quotient ring R/M is a field.^[2] Maximal ideals play a fundamental role in commutative algebra, as every maximal ideal is prime—meaning that if the product of two elements lies in the ideal, then at least one of the elements must belong to it—since fields have no zero divisors.^[1] In any nonzero commutative ring with unity, maximal ideals exist, a result proven using Zorn's lemma on the partially ordered set of proper ideals ordered by inclusion.^[1] For example, in the ring of integers \mathbb{Z}, the principal ideals (p) generated by a prime number p are precisely the maximal ideals.^[2] In principal ideal domains more generally, every nonzero prime ideal is maximal.^[1] The study of maximal ideals is essential for understanding the structure of rings, as they correspond to points in the spectrum of a ring (the set of all prime ideals equipped with the Zariski topology), and they underpin key theorems like Hilbert's Nullstellensatz in algebraic geometry.^[3]

Introduction and Definition

Historical Context

The concept of ideals, which laid the groundwork for maximal ideals, emerged in the late 19th century through Richard Dedekind's efforts to address the failure of unique prime factorization in the rings of algebraic integers. In algebraic number fields, elements often did not factor uniquely into primes, prompting Dedekind to introduce ideals as certain additive subgroups closed under multiplication by ring elements in his 1871 supplement to Dirichlet's Vorlesungen über Zahlentheorie. This innovation allowed him to prove that every nonzero ideal factors uniquely into prime ideals, restoring a structured arithmetic in these domains.^[4] In the early 20th century, David Hilbert advanced ideal theory by applying it to polynomial rings, motivated by problems in invariant theory and algebraic geometry. His 1890 Basis Theorem demonstrated that every ideal in a polynomial ring over a field admits a finite generating set, enabling systematic study of polynomial equations. Complementing this, Hilbert's Nullstellensatz from 1893 established a bijection between maximal ideals in such rings and points in affine space over algebraically closed fields, thus forging a deep connection between algebraic ideals and geometric objects.^[4]^[5] The formalization of commutative ring theory, with maximal ideals gaining prominence as structural cornerstones, occurred in the 1920s and 1930s through contributions by Emmy Noether, Wolfgang Krull, and Emil Artin. Noether's 1921 paper Idealtheorie in Ringbereichen developed an abstract axiomatic framework for ideals in commutative rings, emphasizing ascending chain conditions and primary decomposition, which highlighted the role of maximal ideals in ring classification. Krull, building on this, introduced dimension theory in 1928, defining the Krull dimension via chains of prime ideals terminating at maximal ones, thereby positioning maximal ideals as "points" in the emerging spectral view of rings. Artin extended these ideas to non-commutative settings, but the commutative focus solidified maximal ideals' centrality in abstract algebra. Key milestones include Dedekind's foundational 1871 work and Krull's 1920s advancements in ideal structure.^[6]^[7]^[8]

Formal Definition

In ring theory, a maximal ideal M of a ring R is defined as a proper ideal such that there exists no proper ideal I of R with M \subsetneq I \subsetneq R.^[9] Equivalently, M is maximal if every ideal I containing M satisfies either I = M or I = R.^[10] In commutative rings with unity, an ideal M is maximal if and only if the quotient ring R/M is a field.^[10] For general rings, which may be non-commutative, a two-sided ideal M is maximal if and only if the quotient ring R/M is a simple ring, possessing no nonzero proper two-sided ideals.^[11] Maximal ideals are proper by definition, ensuring they differ from the unit ideal R, and this properness distinguishes their role from prime ideals, though in commutative rings every maximal ideal is prime.^[9]

Examples and Illustrations

In Commutative Rings

In commutative rings, maximal ideals appear prominently in fundamental structures such as fields, the ring of integers, and polynomial rings. A maximal ideal is a proper ideal that is not contained in any larger proper ideal.^[12] Consider a field F, which is a commutative ring where every nonzero element is invertible. The only proper ideal in F is the zero ideal \{0\}, making it the unique maximal ideal, and the quotient ring F/\{0\} \cong F is itself a field.^[13] In the ring of integers \mathbb{Z}, the ideals are principal and of the form n\mathbb{Z} for n \geq 0. The maximal ideals are those generated by prime numbers, i.e., (p) where p is prime, and the quotient \mathbb{Z}/(p) \cong \mathbb{Z}_p (or \mathbb{F}_p) is a field.^[14] For the polynomial ring k over a field k, the ideals are principal, and the maximal ideals are generated by irreducible polynomials, i.e., (f(x)) where f(x) is irreducible over k. If k is algebraically closed, such as the complex numbers \mathbb{C}, then every irreducible polynomial is linear, so the maximal ideals take the form (x - a) for a \in k, corresponding to evaluation at points in k.^[15] In the polynomial ring \mathbb{Z}, maximal ideals include those of the form (p) for prime p, as well as (p, f(x)) where p is prime and f(x) is a polynomial irreducible modulo p. The quotients by these ideals are finite fields. For instance, the ideal (2, x) in \mathbb{Z} is maximal, with quotient \mathbb{Z}/(2, x) \cong \mathbb{Z}/2\mathbb{Z} \cong \mathbb{F}_2.^[16]

In Non-Commutative Rings

In non-commutative rings, maximal ideals are considered separately for left, right, and two-sided cases, extending the commutative notion to one-sided structures where appropriate. A prominent example arises in the full matrix ring M_n(K) over a field K with n \geq 1. This ring is simple, possessing no proper non-zero two-sided ideals, and thus no proper maximal two-sided ideals exist. However, maximal left ideals abound; for each standard basis vector e_j in K^n, the set of all matrices in M_n(K) whose j-th column is the zero vector forms a maximal left ideal, as the quotient module is isomorphic to the simple left M_n(K)-module K^n. Analogous maximal right ideals consist of matrices with a designated row of zeros. In the ring \mathrm{End}_K(V) of all K-linear transformations on a finite-dimensional vector space V, maximal left ideals correspond to one-dimensional subspaces L \subseteq V. Specifically, for each such L, the set \{ T \in \mathrm{End}_K(V) \mid L \subseteq \ker T \} is a maximal left ideal, with the quotient being a simple left module isomorphic to V.^[17] The Weyl algebra A_n(K) over a field K, generated by variables x_1, \dots, x_n and partial derivatives \partial_1, \dots, \partial_n with canonical commutation relations, provides another illustration. As a two-sided ring, it is simple, admitting no non-trivial two-sided ideals. Nonetheless, some of its maximal left ideals are principal (generated by a single element) and intimately linked to the irreducible representations of A_n(K), which are typically infinite-dimensional and holonomic or non-holonomic depending on the generators.^[18] Two-sided maximal ideals also appear in group rings and operator algebras. For the group ring KG over a field K and finite non-abelian group G, the maximal two-sided ideals are the annihilators of the irreducible K-representations of G, yielding quotient rings isomorphic to matrix algebras over division rings. In C*-algebras, the closed two-sided primitive ideals—which serve as the non-commutative analog of maximal ideals—precisely comprise the kernels of irreducible *-representations, with quotients embedding into bounded operators on Hilbert space. Unlike the commutative case, maximal ideals in non-commutative rings need not be prime.

Properties

Basic Properties

A maximal ideal in a ring is proper by definition, meaning it is strictly contained in the ring itself.^[2] The defining property of a maximal ideal M of a ring R ensures that the only ideals of R containing M are M itself and R.^[19] In a commutative ring R with unity, the quotient ring R/[M](/page/M) formed by a maximal ideal M is a field. To see this, consider any nonzero element \overline{a} = a + [M](/page/M) in R/[M](/page/M). The principal ideal generated by \overline{a} in R/[M](/page/M) must be the entire ring R/[M](/page/M), since M is maximal and thus no proper ideal of R strictly contains it, corresponding to no proper ideal in the quotient. Hence, there exists some \overline{b} = b + [M](/page/M) such that \overline{a} \cdot \overline{b} = \overline{1}, providing an inverse for \overline{a}. Every nonzero element thus has a multiplicative inverse, confirming R/[M](/page/M) is a field.^[20] In any ring (not necessarily commutative), the quotient R/[M](/page/M) by a maximal ideal M is a simple ring, meaning it has no proper nonzero two-sided ideals. This follows from the correspondence theorem for ideals, which maps ideals of R containing M bijectively to ideals of R/[M](/page/M); since the only such ideals in R are M and R, the only ideals in R/[M](/page/M) are the zero ideal and R/[M](/page/M) itself.^[21] In commutative rings with unity, finitely generated maximal ideals are principal in certain cases, such as principal ideal domains (PIDs). For instance, in a PID, every maximal ideal is principal, generated by a prime element.^[1]

Relation to Prime Ideals

In commutative rings with identity, every maximal ideal is prime. This follows from the fact that the quotient by a maximal ideal is a field, and every field is an integral domain, so the quotient by a prime ideal must also be an integral domain.^[12]^[22] The converse does not hold: not every prime ideal is maximal. For instance, the zero ideal (0) in the ring of integers \mathbb{Z} is prime, as \mathbb{Z}/(0) \cong \mathbb{Z} is an integral domain, but it is not maximal since \mathbb{Z}/(0) is not a field.^[1] In non-commutative rings, maximal ideals need not be prime. A counterexample arises in the ring T(K, M, K) of triangular matrices over a field K with an infinite-dimensional vector space M as the off-diagonal entry space; the two-sided ideal consisting of matrices with zero in the (1,2)-position is maximal but not prime, as it fails the condition that for ideals A, B with AB \subseteq I, either A \subseteq I or B \subseteq I.^[23] In the spectrum \operatorname{Spec}(R) of a commutative ring R, which consists of all prime ideals equipped with the Zariski topology, the maximal ideals correspond precisely to the closed points. This is because the closure of a maximal ideal \mathfrak{m} is the singleton \{\mathfrak{m}\}, as any prime containing \mathfrak{m} must equal \mathfrak{m}.^[22] The Krull dimension of a ring R, defined as the supremum of the lengths of chains of prime ideals, equals the supremum of the heights of its maximal ideals, where the height of a prime ideal is the length of the longest chain of primes contained in it. Thus, the heights of maximal ideals determine the dimension via such maximal chains ending at maximals.^[24]

Existence Theorems

Zorn's Lemma Approach

To establish the existence of maximal ideals in a unital ring R, consider the partially ordered set \mathcal{P} consisting of all proper ideals of R (i.e., ideals not equal to R), ordered by inclusion \subseteq. This poset is nonempty, as it contains the zero ideal \{0\} assuming R \neq \{0\}.^[25] The poset \mathcal{P} is inductive: every chain in \mathcal{P} has an upper bound in \mathcal{P}. Specifically, for any chain \{I_\alpha\}_{\alpha \in A} of proper ideals, their union I = \bigcup_{\alpha \in A} I_\alpha is an ideal of R. To see that I is proper, suppose toward contradiction that $1 \in I; then $1 belongs to some I_\beta in the chain, implying I_\beta = R, which contradicts the properness of all ideals in the chain. Thus, I is a proper ideal serving as an upper bound.^[26] By Zorn's lemma, since \mathcal{P} is a nonempty inductive poset, it possesses a maximal element M, which is a proper ideal maximal with respect to inclusion among all proper ideals of R. Hence, M is a maximal ideal of R. This argument assumes R has a multiplicative identity $1; without unity, one considers the poset of ideals that do not contain any element acting as a local unit, ensuring properness is preserved in unions.^[27] More generally, for any proper ideal I of the unital ring R, the poset of proper ideals containing I (ordered by inclusion) is also nonempty and inductive, with unions again yielding proper ideals containing I. Applying Zorn's lemma yields a maximal ideal M \supseteq I. Thus, every proper ideal extends to a maximal one.^[25]

Krull's Theorem

Krull's theorem asserts that every nonzero commutative ring with unity has at least one maximal ideal.^[28] This result, established by Wolfgang Krull in 1929, provides a proof of the existence of maximal ideals without relying on the full axiom of choice, instead employing transfinite induction to construct such an ideal.^[28] To show that a maximal ideal contains a given nonzero element a, consider the collection of all prime ideals containing a. Using transfinite induction, build a transfinite chain of prime ideals, each properly containing the previous one and all containing a, continuing until no further proper extension is possible. The union of this chain forms a prime ideal that is maximal among those containing a, and in a commutative ring, this prime ideal is necessarily maximal as an ideal.^[28] Krull's approach generalizes to modules, guaranteeing that every nonzero module over a commutative ring with unity admits a maximal submodule.^[28] While effective for commutative rings, the theorem's method does not directly extend to non-commutative rings, where Zorn's lemma is typically required for existence proofs.^[28]

Applications

Quotient Rings and Fields

In commutative rings with unity, a maximal ideal M yields a quotient ring R/M that is a field.^[20] This field, known as the residue field of R modulo M, captures the structure of R near M and plays a key role in algebraic geometry and number theory by providing a finite or algebraic extension field associated with the ideal.^[29] For instance, in the ring of integers \mathbb{Z}, the principal ideal (5) is maximal, and the quotient \mathbb{Z}/(5) \cong \mathbb{Z}_5 is the finite field with five elements.^[1] In the more general setting of non-commutative rings, a maximal two-sided ideal M produces a quotient R/M that is a simple ring, meaning it possesses no proper nontrivial two-sided ideals.^[30] Simple rings have no nontrivial proper homomorphic images, as any surjective homomorphism from R/M must either be an isomorphism or map to the zero ring.^[31] In the non-commutative case, the elements of R/M do not necessarily form a division ring, but if R/M is left artinian, then by the Wedderburn-Artin theorem, it is isomorphic to a matrix ring M_n(D) over a division ring D.^[32] This structure theorem has significant applications in the classification of simple artinian rings, where maximal ideals correspond to the composition factors of the regular module.^[32] Specifically, in a semisimple artinian ring decomposed as a direct sum of matrix rings over division rings, the maximal two-sided ideals are the annihilators of individual simple components, and the quotients reveal the underlying division rings and their matrix dimensions as the lengths of the composition series for simple modules.^[32]

Spectrum and Local Rings

In ring theory and algebraic geometry, the spectrum of a commutative ring R, denoted \Spec(R), is the set of all prime ideals of R, which serves as the underlying space for the affine scheme associated to R.^[33] Maximal ideals in R correspond to the closed points of \Spec(R), providing a geometric interpretation where these points represent the "classical" points of the associated variety.^[33] The residue field at such a maximal ideal \mathfrak{m}, given by R / \mathfrak{m}, acts as the field of coordinates at that point.^[33] The Zariski topology on \Spec(R) equips this space with a structure where the closed sets are defined as V(I) = \{ \mathfrak{p} \in \Spec(R) \mid I \subseteq \mathfrak{p} \} for any ideal I of R.^[33] This topology has the property that V(I) = V(\sqrt{I}), where \sqrt{I} is the radical of I, ensuring that closed sets correspond bijectively to radical ideals.^[33] Maximal ideals, being prime, appear as points in these closed sets, and the closed points (maximal ideals) are dense in schemes of finite type over a field.^[33] The open sets form a base given by distinguished opens D(f) = \Spec(R) \setminus V((f)) for f \in R, facilitating the study of local properties.^[33] Localization at a maximal ideal \mathfrak{m} of R yields the local ring R_{\mathfrak{m}}, which is a ring with a unique maximal ideal \mathfrak{m} R_{\mathfrak{m}}.^[29] This localization inverts all elements outside \mathfrak{m}, making R_{\mathfrak{m}} a local ring whose residue field is the field of fractions of R / \mathfrak{m}.^[29] Such local rings are essential for studying the behavior of R near the point corresponding to \mathfrak{m} in \Spec(R), as every prime ideal of R_{\mathfrak{m}} is contained in \mathfrak{m} R_{\mathfrak{m}}.^[29] Hilbert's Nullstellensatz establishes a profound connection between algebra and geometry by characterizing maximal ideals in polynomial rings. Specifically, for an algebraically closed field k, every maximal ideal in k[x_1, \dots, x_n] is of the form (x_1 - a_1, \dots, x_n - a_n) for some (a_1, \dots, a_n) \in k^n.^[15] This weak form implies a bijection between maximal ideals and points in affine n-space over k, ensuring that proper ideals have common zeros.^[15] The strong form further states that if a polynomial g vanishes on the zero set of an ideal generated by f_1, \dots, f_m, then some power g^r lies in that ideal.^[34] The Krull dimension of R is defined as the supremum of the lengths of chains of strictly increasing prime ideals in \Spec(R), where the length is the number of strict inclusions.^[35] In saturated chains, the final prime is maximal, so the dimension equals the supremum of the heights of maximal ideals, measuring the "size" of the space \Spec(R).^[35] This dimension coincides with the transcendence degree of the fraction field of R when R is the coordinate ring of an irreducible affine variety.^[35] Hilbert's Nullstellensatz links algebraic geometry to commutative algebra by providing a correspondence between geometric varieties (zero sets of ideals) and algebraic ideals in polynomial rings over algebraically closed fields.^[36] It enables the translation of geometric problems, such as the emptiness of varieties, into algebraic conditions on ideals, facilitating proofs of existence or non-existence of solutions to polynomial systems.^[36] This bridge underpins much of modern algebraic geometry, allowing algebraic tools to resolve geometric questions about affine spaces.^[36]

Generalizations

To Modules

In module theory, the concept of a maximal ideal in a ring generalizes to that of a maximal submodule in a module. A maximal submodule N of an R-module M is a proper submodule (i.e., N \neq M) such that there exists no submodule K with N \subsetneq K \subsetneq M.^[37] This condition is equivalent to the quotient module M/N being simple, meaning M/N has no proper nonzero submodules.^[38] Simple modules serve as the basic building blocks in the structure theory of modules, analogous to simple rings or fields in ring theory. Over a unital ring R, not every nonzero module necessarily possesses a maximal submodule; counterexamples include the rational numbers \mathbb{Q} viewed as a \mathbb{Z}-module, where proper submodules form an ascending chain without a maximum. However, for every nonzero finitely generated R-module M, a maximal submodule exists and can be established using Zorn's lemma on the partially ordered set of proper submodules of M, which is nonempty and inductive under the finite generation assumption.^[39] A notable special case arises when M is a vector space over a field k, in which case the maximal submodules are precisely the hyperplanes—subspaces of codimension 1.^[40] These correspond to the kernels of nonzero linear functionals on M. When the ring R is regarded as a left R-module _RR, its submodules are exactly the left ideals of R, so the maximal submodules of _RR are the maximal left ideals of R.^[41]

To Other Algebraic Structures

In commutative algebras over an algebraically closed field, maximal ideals correspond to the kernels of one-dimensional irreducible representations, which arise as evaluation homomorphisms at points in the associated affine variety.^[42] In commutative C*-algebras, maximal ideals are precisely the kernels of irreducible *-representations, which are one-dimensional and correspond to evaluation at points in the underlying compact Hausdorff space via the Gelfand transform.^[43] In lattice theory, a maximal ideal is a maximal proper ideal, defined as a down-set that is maximal among proper down-sets closed under finite joins, and such ideals are necessarily prime in distributive lattices.^[44] In partially ordered sets, maximal ideals extend this notion to maximal proper down-sets, playing a role analogous to maximal filters in the order dual.^[44] In category theory, within an abelian category, a maximal subobject of an object is a proper monomorphism whose cokernel is a simple object, meaning the quotient has no nontrivial subobjects.^[45] The concept of maximal ideals generalizes to semigroups, where a maximal ideal is a maximal proper left (or two-sided) ideal, and in finite semigroups, every proper subsemigroup is contained in a maximal proper subsemigroup.^[46]

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