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Wedderburn–Artin theorem

The Wedderburn–Artin theorem states that every simple left Artinian ring is isomorphic to a matrix ring over a division ring, and more generally, every semisimple Artinian ring is isomorphic to a finite direct product of such matrix rings.^[1] This classification provides a complete structural description of these rings, revealing them as direct sums of matrix algebras M_{n_i}(D_i), where each D_i is a division ring and n_i is a positive integer.^[2] The theorem originated with Joseph H. M. Wedderburn's 1908 work on hypercomplex numbers, where he established the result for finite-dimensional semisimple algebras over fields.^[3] In 1927, Emil Artin extended it to the broader class of semisimple rings satisfying the descending chain condition on left ideals, introducing key concepts like Artinian rings in the process.^[2] This generalization solidified the theorem's role in noncommutative ring theory, building on earlier ideas from Dickson and others on linear associative algebras. The theorem has profound implications for representation theory and algebra, implying that semisimple rings are Morita equivalent to products of division rings and enabling the study of their modules as direct sums of matrix modules.^[2] It also underpins results like the density theorem for primitive rings and connects to group algebras, where semisimple group rings over fields of characteristic zero decompose accordingly. Furthermore, it distinguishes commutative cases, where the division rings are fields, leading to Artin–Wedderburn decompositions for semisimple commutative rings as products of matrix rings over fields.^[2]

Preliminaries

Artinian Rings

A ring R is left Artinian if every descending chain of left ideals stabilizes, meaning that for any sequence I_1 \supseteq I_2 \supseteq I_3 \supseteq \cdots of left ideals in R, there exists an integer n such that I_k = I_n for all k \geq n.^[4] This condition is equivalent to every nonempty collection of left ideals in R containing a minimal element with respect to inclusion.^[5] The descending chain condition on left ideals is further equivalent to R having finite length as a left module over itself.^[6] In this characterization, the length of R is the length of a composition series for R as a left R-module, consisting of simple subquotients. Examples of left Artinian rings include finite direct products of full matrix rings over division rings.^[7] For instance, the ring of n \times n matrices over a division ring D satisfies the descending chain condition on left ideals because it has finite length n as a left module over itself. In the commutative case, Artinian principal ideal domains are precisely fields, since any nonzero proper ideal would generate an infinite descending chain otherwise.^[8] Left Noetherian rings satisfy the ascending chain condition on left ideals, requiring that every ascending chain of left ideals stabilizes.^[9] This contrasts with the descending chain condition defining left Artinian rings, though the two conditions coincide for modules of finite length. Indeed, every left Artinian ring is left Noetherian, as the finite length of R over itself implies both chain conditions hold for left submodules.^[6] A key property of left Artinian rings is that there are only finitely many simple left modules up to isomorphism.^[10] This follows from the existence of a composition series for R as a left module over itself, whose factors are simple modules, and any two composition series have the same simple factors up to isomorphism and multiplicity by the Jordan-Hölder theorem.

Semisimple Rings

A left semisimple ring R is defined as a ring in which every left R-module is semisimple, meaning it decomposes as a direct sum of simple left R-modules. Equivalently, R itself is semisimple as a left module over itself, i.e., R \cong \bigoplus_{i \in I} S_i for some index set I and simple left R-modules S_i. This module-theoretic characterization emphasizes the complete decomposability property that underpins the structure of such rings.^[11]^[12] Several conditions are equivalent to this definition. For instance, R is left semisimple if and only if every left R-module is both injective and projective. Additionally, in the context of Artinian rings, left semisimplicity is equivalent to the Jacobson radical J(R) = 0, where the Artinian condition ensures finite-length decompositions into simple modules.^[11]^[13] Examples of semisimple rings include full matrix rings M_n(D) over a division ring D, as these decompose into simple modules corresponding to the standard representation. More generally, finite direct products of such matrix rings, like M_{n_1}(D_1) \times \cdots \times M_{n_k}(D_k), are also semisimple, illustrating how the structure builds from basic building blocks. Over fields, full matrix rings M_n(K) for a field K provide concrete finite-dimensional instances.^[14] A key property of semisimple rings is that the socle of R, defined as the sum of all simple left submodules of R, coincides with R itself, reflecting the absence of non-semisimple components. This equality underscores the ring's full decomposability and serves as a prerequisite for deeper structural results in ring theory.^[11]

Simple Modules and Schur's Lemma

A simple left R-module is defined as a nonzero left R-module M that admits no proper nonzero submodules. This means that the only submodules of M are \{0\} and M itself, making simple modules the "indecomposable building blocks" in the category of left R-modules. A key property of simple modules is that any R-module homomorphism f: M \to N between simple modules M and N is either the zero map or an isomorphism. To see this, note that if f \neq 0, then \ker f is a proper submodule of M (neither \{0\} nor M), which is impossible unless \ker f = \{0\}, so f is injective; similarly, \operatorname{im} f is a nonzero submodule of N, hence \operatorname{im} f = N, so f is surjective. In particular, if M \not\cong N, then \operatorname{Hom}_R(M, N) = \{0\}. Schur's lemma provides a fundamental characterization of the endomorphism ring of a simple module. Specifically, for a simple left R-module M, the endomorphism ring \operatorname{End}_R(M) = \operatorname{Hom}_R(M, M) is a division ring. The proof follows directly from the property above: for any nonzero f \in \operatorname{End}_R(M), the map f is an isomorphism (hence invertible in \operatorname{End}_R(M)), since both \ker f = \{0\} and \operatorname{im} f = M as submodules of the simple module M. Thus, every nonzero element of \operatorname{End}_R(M) has a multiplicative inverse, establishing its division ring structure. This result, originally due to Issai Schur in the context of group representations and later generalized to modules over rings, underscores the rigid nature of simple modules under endomorphisms. In notation, if M is a simple left R-module, then \operatorname{End}_R(M) \cong D for some division ring D. A concrete example illustrates this: let k be a field and R = M_n(k) the ring of n \times n matrices over k. The standard left R-module is the column vector space k^n, which is simple since any nonzero R-submodule would be the entire space (as matrix actions preserve linear independence in this context). Here, \operatorname{End}_R(k^n) \cong k, which is a field (hence a division ring), consisting precisely of scalar multiplications by elements of k.

Theorem Statement

General Form

The Wedderburn–Artin theorem describes the structure of semisimple Artinian rings. It asserts that every left Artinian semisimple ring R is isomorphic to a finite direct product R \cong \prod_{i=1}^k M_{n_i}(D_i), where each D_i is a division ring and each n_i \geq 1 is a positive integer.^[2]^[15] Here, M_n(D) denotes the ring of n \times n matrices over the division ring D, equipped with the standard matrix addition and multiplication.^[2] The product is finite because the left Artinian condition on R implies a descending chain condition on left ideals, ruling out infinite direct sums.^[15] Moreover, the division rings D_i are unique up to isomorphism, and the positive integers n_i are unique, both up to permutation of the indices.^[15]

Case for Finite-Dimensional Algebras

When the base ring is a field k and the semisimple algebra A is finite-dimensional over k, the Wedderburn–Artin theorem provides a refined structure theorem that leverages the vector space properties of A. In this setting, the general decomposition into matrix rings over division rings simplifies due to the finite dimensionality, ensuring all components are finite-dimensional as well.^[16] Specifically, a finite-dimensional semisimple k-algebra A is isomorphic to a direct product \prod_{i=1}^r M_{n_i}(D_i), where each D_i is a finite-dimensional division k-algebra and each n_i is a positive integer. This isomorphism arises from the decomposition of A into simple components, each corresponding to a matrix algebra over the endomorphism ring of a simple module, which turns out to be a division algebra.^[16] The decomposition is unique up to isomorphism and permutation of factors: the integers n_i are determined by the multiplicities of the simple modules, while the division algebras D_i are unique up to isomorphism. Invariants include the centers Z(D_i), which are field extensions of k, and the indices [D_i : Z(D_i)], the dimensions of D_i over their centers, which classify the simple components up to Brauer equivalence over Z(D_i).^[17] If A is moreover a central simple k-algebra—meaning its center is exactly k and it has no nontrivial two-sided ideals—then the decomposition simplifies further to A \cong M_n(D), where D is a central division k-algebra (so Z(D) = k) and n \geq 1 is the matrix size such that the degree of A (defined as \sqrt{\dim_k A}) equals n times the degree of D (or the index of A, \sqrt{[D : k]}). In this case, the dimension [D : k] is a square.^[18] An illustrative example occurs over the real numbers k = \mathbb{[R](/page/R)}, where non-commutative division algebras like the Hamilton quaternions \mathbb{H} (with \dim_{\mathbb{R}} \mathbb{H} = 4) appear as factors D_i; for instance, the algebra \mathbb{H} itself is a central simple \mathbb{R}-algebra isomorphic to M_1(\mathbb{H}), while split forms like M_2(\mathbb{R}) correspond to trivial division factors.^[18]

Proof Outline

Module Decomposition

In the proof of the Wedderburn–Artin theorem, the initial step involves decomposing the right regular module R_R of an Artinian semisimple ring R. Since R is semisimple, every right R-module, including R_R, is semisimple, meaning every submodule is a direct summand.^[19] Moreover, as R is Artinian, R_R has finite length, ensuring it admits a decomposition as a finite direct sum of simple right R-modules.^[15] Specifically, there exists a finite collection of pairwise non-isomorphic simple right R-modules I_1, \dots, I_m and positive integers n_1, \dots, n_m such that

R_R \cong \bigoplus_{i=1}^m I_i^{\oplus n_i}

as right R-modules.^[20] Each simple summand I_i corresponds to a minimal right ideal of R, and the multiplicities n_i reflect the dimension of the corresponding isotypic component in the decomposition.^[15] This decomposition is unique up to isomorphism of the simple modules and permutation of the summands, a consequence of the Krull–Schmidt theorem, which applies to Artinian modules of finite length such as those over semisimple Artinian rings.^[21] The theorem guarantees that any two direct sum decompositions into indecomposable summands are equivalent, providing the uniqueness essential for the structural analysis of R.^[21] In the homological context, right ideals of R serve as submodules of R_R, and the semisimplicity of R ensures that every such submodule—and in particular, every minimal right ideal—is a direct summand.^[19] This property facilitates the explicit construction of the decomposition using idempotents. A set of pairwise orthogonal primitive idempotents e_1, \dots, e_s in R (summing to the identity) can be selected such that

R = \bigoplus_{j=1}^s e_j R

as right R-modules, where each e_j R is a simple right ideal (hence isomorphic to one of the I_i).^[15] The primitivity of the e_j ensures the indecomposability of these summands, aligning with the simple module structure.^[20]

Endomorphism Rings and Division Structure

Following the module decomposition of the regular left module into a direct sum of its simple submodules, the proof proceeds by determining the structure of the ring through the endomorphism rings of these summands. For each simple left R-submodule I_i in the decomposition, Schur's lemma implies that the endomorphism ring \mathrm{End}_R(I_i) \cong D_i, where D_i is a division ring.^[22] The simple submodules I_i and I_j for i \neq j are non-isomorphic, so the Hom-spaces satisfy \mathrm{Hom}_R(I_i, I_j) = 0.^[23] When a given simple module appears with multiplicity in the decomposition, the corresponding isotypic component V_i (the direct sum of all submodules isomorphic to I_i) has the form V_i \cong I_i^{\oplus m_i} for some integer m_i \geq 1, and this multiplicity determines the matrix size in the ring structure.^[15] Specifically, \mathrm{End}_R(V_i) \cong M_{n_i}(D_i), where the integer n_i accounts for the multiplicity such that \dim_{D_i} \mathrm{Hom}_R(V_i, V_i) = n_i^2.^[24] The left R-action on the regular module

\, _RR \cong \bigoplus V_i

preserves each isotypic component V_i, since the Hom-spaces between distinct components vanish. This induces a ring homomorphism R \to \bigoplus_i \mathrm{End}_R(V_i). The Peirce decomposition of R with respect to the orthogonal primitive idempotents projecting onto the components V_i reveals the block structure, where the opposite ring R^\mathrm{op} acts on each V_i via right multiplication.^[23] By the double centralizer theorem, the image of each block of R in \mathrm{End}_R(V_i) is the full matrix ring, yielding the isomorphism of the corresponding summand with M_{n_i}(D_i).^[22] Combining the summands across all components gives the full ring isomorphism R \cong \bigoplus_i M_{n_i}(D_i).^[15] This establishes the matrix-over-division-ring form, completing the structural description of the semisimple Artinian ring.^[24]

Consequences

Simple Algebras

A simple Artinian ring R, meaning it has no nontrivial two-sided ideals and satisfies the descending chain condition on left ideals, is semisimple and thus has zero Jacobson radical. As a special case of the Wedderburn–Artin theorem with a single summand, such a ring is isomorphic to a full matrix ring M_n(D) over a division ring D, where n \geq 1 is uniquely determined by the dimension of the unique simple left R-module over D. The proof proceeds by first noting that since R is left Artinian and simple, it admits a minimal left ideal L, and because R is prime, L^2 \neq 0. This yields a primitive idempotent e such that eRe is a division ring D, and the left ideal Re is simple. The regular module {}_R R then decomposes as a direct sum of n copies of Re, where n = [R : Re], the index of Re in R. By the double centralizer theorem or Schur's lemma applied to the endomorphism ring, this forces R \cong M_n(D).^[1] Moreover, since R is semisimple, its Jacobson radical J(R) = 0, and the semisimple quotient R/J(R) being simple aligns directly with the division ring matrix form without needing reduction. A concrete example is the ring of m \times m matrices over a field F, which is a simple Artinian ring isomorphic to M_m(F) with D = F and n = m, exhibiting the full structure as semisimple with a unique simple module up to isomorphism.

Central Simple Algebras

A central simple k-algebra, where k is a field, is defined as a finite-dimensional algebra over k that is simple (possessing no nontrivial two-sided ideals) and central (having center precisely equal to k).^[25] The Wedderburn–Artin theorem yields a fundamental corollary for these algebras: every finite-dimensional central simple k-algebra A is isomorphic to M_n(D), where n \geq 1 is an integer and D is a central division k-algebra (a division algebra with center k).^[25] Here, n is uniquely determined as the unique integer such that the simple modules over A have dimension n over D, and D is unique up to k-isomorphism.^[25] The dimension relation follows immediately: if [A : k] = \dim_k A, then [A : k] = n^2 [D : k].^[25] The index of A, denoted \mathrm{ind}(A), is defined as the degree of the central division algebra D in this decomposition, specifically \mathrm{ind}(A) = \sqrt{[D : k]}.^[25] Thus, [A : k] = n^2 \cdot \mathrm{ind}(A)^2. The period of A, denoted \mathrm{per}(A), is the multiplicative order of the Brauer class [A] in the Brauer group \mathrm{Br}(k). A key arithmetic relation is that \mathrm{per}(A) divides \mathrm{ind}(A), and moreover, \mathrm{per}(A) and \mathrm{ind}(A) share the same prime divisors.^[25] Classification of central simple k-algebras proceeds via the Brauer group \mathrm{Br}(k), which is the torsion abelian group of equivalence classes of central simple k-algebras under the relation of Brauer equivalence: two such algebras A and B are equivalent if A \otimes_k B^{\mathrm{op}} \cong M_m(E) for some integer m \geq 1 and some central simple k-algebra E.^[25] The group operation is induced by the tensor product over k. The Wedderburn–Artin theorem implies that each element of \mathrm{Br}(k) corresponds precisely to a (unique up to isomorphism) central division k-algebra, modulo matrix algebra envelopes; that is, the class [A] is represented by the division kernel D in the decomposition A \cong M_n(D).^[25] Emil Artin's contributions to the theory of noncommutative rings, including his 1920s work on Artinian rings and 1940s investigations into semisimple algebras over the rationals, connected the structural decomposition from the Wedderburn–Artin theorem to cohomological invariants in the Brauer group, notably through his role in establishing the local-global principle for central simple algebras over number fields (jointly with Brauer and Hasse).^[26]

Over Algebraically Closed Fields

When the base field k is algebraically closed, the Wedderburn–Artin theorem admits a particularly simple form for finite-dimensional semisimple k-algebras. Specifically, every such algebra A is isomorphic to a direct sum of matrix algebras over k:

A \cong \bigoplus_{i=1}^r M_{n_i}(k),

where each n_i \geq 1 is an integer and r is the number of simple components. This follows directly from the general structure theorem, as the only finite-dimensional division algebra over an algebraically closed field is k itself.^[27] The absence of nontrivial finite-dimensional division k-algebras over algebraically closed k stems from the fact that any element x \in D (for a division algebra D) satisfies a polynomial equation over k, and since k has no proper algebraic extensions, x must lie in k, forcing D = k. This trivialization simplifies the decomposition, reducing the general case of matrix rings over division rings to pure matrix rings over the field.^[27]^[28] In representation theory, this corollary implies that semisimple algebras over the complex numbers \mathbb{C} (which is algebraically closed) decompose into blocks of full matrix algebras, each corresponding to the endomorphism ring of an irreducible representation. This structure facilitates the classification of representations, as the simple modules are precisely the standard modules over these matrix algebras. A prominent example arises with group algebras of finite groups. By Maschke's theorem, the group algebra \mathbb{C}[G] of a finite group G is semisimple, and thus

\mathbb{C}[G] \cong \bigoplus_{i=1}^r M_{n_i}(\mathbb{C}),

where the sum runs over the irreducible representations of G and each n_i is the dimension of the i-th irreducible representation. This decomposition underscores the theorem's role in character theory and the orthogonality relations for representations.^[29]

Historical Context

Wedderburn's Work

Joseph Wedderburn (1882–1948) was a Scottish mathematician renowned for his pioneering contributions to abstract algebra, particularly in the theory of associative algebras and ring structures. Born in Forfar, Scotland, he studied at the University of Edinburgh and Princeton University, where he later held a professorial position for much of his career. Wedderburn's work bridged classical hypercomplex number systems, such as quaternions and Clifford algebras, with modern abstract approaches to algebras over fields.^[30] In his landmark 1908 paper "On Hypercomplex Numbers," published in the Proceedings of the London Mathematical Society, Wedderburn established foundational results on the structure of finite-dimensional semisimple associative algebras, referred to as hypercomplex systems.^[3] He demonstrated that a semisimple algebra without nilpotent ideals decomposes uniquely as a direct sum of simple algebras. Each simple algebra is isomorphic to a matrix ring over a division algebra finite-dimensional over the base field. Wedderburn explicitly included the quaternion algebra as an example of a primitive division algebra that is not a field, allowing for non-commutative components in the decomposition. These results handled the finite-dimensional case over arbitrary fields. Wedderburn's theorems emphasized the role of invariant subalgebras and normal series in determining the indecomposable components, proving that maximal nilpotent invariant subalgebras exist and that semisimple algebras are precisely those with no non-trivial nilpotent ideals. However, his framework did not fully address the classification of primitive algebras over general division rings. These limitations left open the general Artinian case beyond finite-dimensional algebras over fields. Wedderburn's 1908 work laid the essential groundwork for the full Wedderburn–Artin theorem by providing the decomposition for semisimple finite-dimensional algebras and highlighting the centrality of division rings in simple components. His insights influenced subsequent generalizations, establishing matrix rings over division algebras as the building blocks of semisimple structures.

Artin's Contributions

Emil Artin (1898–1962) was an Austrian mathematician renowned for his work in algebra, particularly in ring theory and Galois theory. Born in Vienna, he studied at the University of Leipzig and later held positions at the University of Göttingen and the University of Hamburg, where he made significant advances in noncommutative algebra during the 1920s.^[26] In 1927, Artin published the seminal paper "Zur Theorie der hypercomplexen Zahlen" in the Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, which provided a groundbreaking generalization of Wedderburn's earlier results on finite-dimensional semisimple algebras. Building upon Wedderburn's 1908 work, Artin extended the decomposition to arbitrary semisimple Artinian rings, showing that such rings are isomorphic to direct products of matrix rings over division rings, without restricting to algebras over fields. This advancement relied on a module-theoretic approach, treating the ring as acting on itself as a module and leveraging the Artinian condition to ensure finite-length modules and semisimple structure.^[26]^[2] Artin's proof established that the endomorphism rings of minimal ideals are division rings and introduced the double centralizer theorem to prove the uniqueness of the decomposition, using centralizers to identify invariant substructures within the ring. This theorem states that for a semisimple subalgebra acting on a module, the centralizer of its centralizer recovers the original algebra, providing a key tool for structural analysis.^[31]^[2] Artin's contributions completed the Wedderburn–Artin theorem, solidifying its role as a cornerstone of noncommutative ring theory and influencing subsequent developments, such as Richard Brauer's work on the Brauer group, which classifies central simple algebras up to Morita equivalence using the theorem's matrix-division ring decomposition.^[26]

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