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Semisimple module

In ring theory, a semisimple module over a ring R is an R-module that decomposes as a direct sum of simple R-modules, where simple modules are nonzero modules with no proper nonzero submodules.^[1]^[2] This decomposition may be finite or infinite, though in many contexts—such as finitely generated modules—it is finite.^[1] Semisimplicity admits several equivalent characterizations: a module M is semisimple if and only if it is a sum (not necessarily direct) of simple submodules; if every submodule of M is a direct summand; or if every surjective homomorphism from M onto another module splits.^[2]^[1] These properties highlight the "reducibility" of semisimple modules, meaning they break down completely into irreducible building blocks without further complications from extensions.^[2] Key properties of semisimple modules include the uniqueness of their simple summands up to isomorphism and permutation; the fact that all submodules and quotient modules of a semisimple module are themselves semisimple; and, for Artinian modules, the condition that the Jacobson radical vanishes.^[1] Finite-length semisimple modules are both Noetherian and Artinian, ensuring well-behaved ascending and descending chains of submodules.^[1] In representation theory, semisimple modules play a central role, corresponding to completely reducible representations of algebras or groups; for instance, over fields of characteristic zero, representations of finite groups are semisimple by Maschke's theorem, decomposing into direct sums of irreducibles.^[3] A ring is called semisimple if it is semisimple as a module over itself, leading to structure theorems like the Artin-Wedderburn theorem, which describes such rings as products of matrix algebras over division rings.^[4]

Fundamental Concepts

Simple modules

A simple module over a ring R is defined as a nonzero R-module M that admits no proper nontrivial submodules; that is, the only submodules of M are the zero submodule \{0\} and M itself. This condition captures the notion of an "atomic" or indecomposable unit in module theory, where M cannot be broken down further into smaller substructures under the ring action. Equivalently, every nonzero element of M generates the entire module as a cyclic submodule, ensuring that the module is as minimal as possible while being nonzero.^[5] Classic examples of simple modules illustrate this minimality across different rings. Over a field k, any one-dimensional vector space is a simple k-module, as its only subspaces are \{0\} and itself. For the ring \mathbb{Z} of integers, the cyclic group \mathbb{Z}/p\mathbb{Z} of prime order p forms a simple \mathbb{Z}-module, since its only subgroups are the trivial ones. In the context of group representation theory, an irreducible representation of a group G over a field corresponds to a simple module over the group algebra k[G], where no proper invariant subspaces exist under the group action.^[5]^[6]^[7] The simplicity of such modules underscores their role as indecomposable building blocks, distinct from broader decompositions into direct sums. Basic properties follow from the submodule lattice: by the correspondence theorem, submodules of a quotient module M/N align bijectively with submodules of M containing N, which implies that every nonzero module has at least one simple quotient (obtained by modding out a maximal proper submodule). This property highlights simplicity as a form of extreme indecomposability at the level of submodules, without requiring further structural assumptions on the module.^[8] The concept of simple modules draws an analogy to simple groups in group theory, where no nontrivial normal subgroups exist, and was introduced in the development of ring theory during the 1920s by Emmy Noether, particularly in her work on ideals and noncommutative structures.^[9]^[10]

Direct sums and summands

The direct sum of a family of modules \{M_i\}_{i \in I} over a ring R, also known as the external direct sum and denoted \bigoplus_{i \in I} M_i, consists of all families (m_i)_{i \in I} with m_i \in M_i and only finitely many m_i nonzero, equipped with componentwise addition and scalar multiplication.^[8] This construction ensures that each M_i embeds naturally as a submodule via the map sending m \in M_i to the family with m in the i-th position and zeros elsewhere, and the sum of these images is direct.^[8] An internal direct sum arises when a module M is expressed as the sum of submodules N_j \subseteq M (j \in J) such that every element of M is a finite sum of elements from the N_j and the intersection of any N_k with the sum of the others is zero.^[8] In this case, M \cong \bigoplus_{j \in J} N_j as R-modules, with the isomorphism identifying each element uniquely as such a finite sum.^[8] A submodule N \subseteq M is a direct summand if there exists another submodule K \subseteq M such that M is the internal direct sum of N and K.^[8] Direct sums of modules preserve exactness of sequences in the category of R-modules, meaning that if $0 \to A_i \to B_i \to C_i \to 0is exact for eachi

, then &#36;0 \to \bigoplus_i A_i \to \bigoplus_i B_i \to \bigoplus_i C_i \to 0

is also exact.^[11] This property holds because direct sums are both products and coproducts (biproducts) in this abelian category, facilitating homological computations.^[11] Idempotents in the endomorphism ring \operatorname{End}_R(M) of a module M correspond precisely to direct sum decompositions of M. Specifically, if e \in \operatorname{End}_R(M) satisfies e^2 = e, then M = \operatorname{im}(e) \oplus \ker(e), where \operatorname{im}(e) is the image of e and \ker(e) is its kernel.^[12] Conversely, given a decomposition M = N \oplus K, the projection onto N along K defines an idempotent endomorphism with image N and kernel K.^[12] This bijection between nontrivial proper idempotents and nontrivial direct sum decompositions underscores the structural role of direct summands. While external direct sums over infinite index sets require only finitely many nonzero components, finite direct sums suffice for many module-theoretic properties, such as semisimplicity in settings without artinian assumptions, where infinite sums may introduce complications like non-noetherian behavior.^[8]

Definition and Characterizations

Primary definition

In ring theory, a module M over a ring R (with identity) is defined to be semisimple if every submodule of M is a direct summand of M.^[4] This means that for any submodule N \subseteq M, there exists a submodule P \subseteq M such that M = N \oplus P.^[13] An equivalent characterization is that M is semisimple if it is isomorphic to a direct sum of simple R-modules, where the sum may be infinite.^[13] In this case, M \cong \bigoplus_{i \in I} S_i, with each S_i a simple module (i.e., having no proper nonzero submodules).^[4] Basic examples include finite-dimensional vector spaces over a field k, viewed as k-modules; here, every subspace is a direct summand via a choice of basis complement.^[13] Another example arises in representation theory: the group algebra \mathbb{C}[G] for a finite group G is semisimple as a left module over itself, by Maschke's theorem, since the characteristic zero ensures every submodule is a summand.^[13] The two definitions are related as follows: assuming every submodule is a summand, the collection of all direct sums of simple submodules of M is partially ordered by inclusion; by Zorn's lemma, a maximal such sum exists, and it equals M because any proper containment would allow extension by a simple summand generated from a nonzero element outside it.^[14] The converse holds by properties of direct sums, where submodules are themselves direct sums of simples and thus summands.^[4]

Equivalent conditions

A module M over a ring R is semisimple if and only if every submodule of M is a direct summand.^[15] This condition is equivalent to M being expressible as a direct sum of simple submodules, which in turn is equivalent to M being a sum (not necessarily direct) of simple submodules.^[15]^[16] To see that the sum of simple submodules equals M implies a direct sum decomposition, consider a maximal direct sum \bigoplus_{i \in I} S_i of simple submodules such that its intersection with the remaining sum is zero; any additional simple submodule must then be absorbed into this direct sum without overlap, yielding the full decomposition.^[15] Conversely, if every submodule is a direct summand, then starting from zero, iteratively extracting simple submodules (which exist in any nonzero submodule) and their complements covers M as a direct sum.^[15] For modules of finite length, semisimplicity is equivalent to the existence of a composition series with simple factors in which every short exact sequence arising from consecutive terms splits.^[17] In such cases, the Jordan-Hölder theorem ensures all composition series have the same length and isomorphic factors up to permutation, and the splitting property follows from the direct sum decomposition into simples, as each step in the series corresponds to a direct summand.^[17] More generally, M is semisimple if and only if every short exact sequence $0 \to N \to M \to Q \to 0 with N and Q semisimple splits, since submodules and quotients of semisimple modules inherit semisimplicity, allowing inductive decomposition.^[18] Another characterization is that the socle \soc(M), the sum of all simple submodules of M, equals M.^[16] This holds because the socle is the largest semisimple submodule, and if it coincides with M, then M is semisimple by the sum-of-simples condition.^[16] Dually, for finitely generated modules over finite-dimensional algebras, semisimplicity is equivalent to the radical \rad(M) = 0, where the radical is the intersection of all maximal submodules (or kernels of epimorphisms onto simples).^[16] The equivalence arises because a nonzero radical would yield a proper essential submodule (intersecting every simple submodule nontrivially), contradicting the direct summand property of submodules in semisimple modules.^[16] In certain ring classes, such as semisimple artinian rings, a module M is semisimple if and only if it is both projective and injective.^[2] Over such rings, all modules are semisimple, hence both projective (as direct summands of frees) and injective (via splitting of monomorphisms).^[2] However, this equivalence fails in general; for instance, over the integers \mathbb{Z}, the module \mathbb{Z}/p\mathbb{Z} (for prime p) is simple, hence semisimple, but neither projective (as projectives over \mathbb{Z} are free) nor injective (as injectives are divisible groups).^[2] Over commutative rings, semisimplicity coincides with the module being completely reducible, meaning every submodule is complemented by a direct summand.^[15] For finitely generated modules of finite length, the Krull-Schmidt theorem provides uniqueness of the direct sum decomposition up to isomorphism and multiplicity of indecomposable (simple) summands, relying on the endomorphism rings of simples being division rings by Schur's lemma.^[16] The submodule summand property ensures no proper essential submodules exist, as any essential N \subsetneq M would intersect its complement trivially, violating essentiality.^[15]

Structural Properties

Decomposition theorems

A fundamental result in the theory of semisimple modules is the existence of a direct sum decomposition into simple submodules. Specifically, every semisimple left R-module M is isomorphic to a direct sum of simple submodules.^[14] To prove this, consider the set of all families of simple submodules whose direct sum is a submodule of M; this set is nonempty since it contains the empty family and single simple submodules. By Zorn's lemma, there exists a maximal such family \{S_\alpha\}_{\alpha \in I}. If the direct sum \bigoplus_{\alpha \in I} S_\alpha \neq M, then there is a simple submodule S \subseteq M not contained in this sum, and since M is semisimple, S complements the sum as a direct summand, contradicting maximality. Thus, M \cong \bigoplus_{\alpha \in I} S_\alpha.^[14] An alternative proof uses transfinite induction on the partial order of direct sums of simples, extending maximal decompositions at limit ordinals.^[2] Decompositions of semisimple modules may be finite or infinite. If M is artinian and semisimple, then its decomposition into simple summands is finite.^[1] This follows because an infinite direct sum of nonzero simples would admit a strictly descending chain of submodules given by the partial sums, violating the descending chain condition. In contrast, infinite decompositions exist; for example, over the ring \mathbb{Z} of integers, the infinite direct sum \bigoplus_{n=1}^\infty \mathbb{Z}/p\mathbb{Z} (for a fixed prime p) is semisimple, as each \mathbb{Z}/p\mathbb{Z} is a simple \mathbb{Z}-module and direct sums preserve semisimplicity, but it is not artinian.^[2] The simple summands in a decomposition of a semisimple module M can be grouped by isomorphism classes into isotypic components. For each isomorphism class of simple modules represented by V, the V-isotypic component of M is the direct sum of all submodules of M isomorphic to V, which takes the form V \otimes_k k_V, where k_V is a vector space over the division ring \operatorname{End}_R(V)^{\mathrm{op}} with dimension equal to the multiplicity of V in M, and M is the direct sum over all such distinct simples V of these components.^[1] For semisimple modules M = \bigoplus_i S_i^{n_i} and N = \bigoplus_j T_j^{m_j} (with S_i, T_j simple), the spaces \operatorname{Hom}_R(M, N) and \operatorname{Ext}^1_R(M, N) decompose according to the simple components, as both functors respect direct sums: \operatorname{Hom}_R(M, N) \cong \bigoplus_{i,j} \operatorname{Hom}_R(S_i^{n_i}, T_j^{m_j}) and similarly for \operatorname{Ext}^1_R(M, N), with nonzero terms only when S_i \cong T_j.^[2] In particular, for the endomorphism ring, if M = \bigoplus_i S_i^{n_i} with each \operatorname{End}_R(S_i) a division ring D_i, then \dim \operatorname{Hom}_R(M, M) = \sum_i n_i^2 \dim D_i.^[1]

\operatorname{End}_R(M) \cong \prod_i M_{n_i}(D_i), \quad \dim \operatorname{End}_R(M) = \sum_i n_i^2 \dim D_i

Uniqueness and multiplicities

For a semisimple module M of finite length over a ring R, the Krull–Schmidt–Azumaya theorem guarantees that any direct sum decomposition M \cong \bigoplus_i S_i^{n_i} into simple summands S_i (up to isomorphism) is unique up to isomorphism of the summands and permutation of the indices.^[19] This uniqueness extends to general semisimple modules through their isotypic decomposition, where the isotypic component corresponding to a simple module S—the direct sum of all summands isomorphic to S—is uniquely determined as the maximal submodule on which the endomorphism ring acts via the division ring \End_R(S).^[20] The multiplicity n_i of a simple summand S_i in the decomposition is an invariant given by n_i = \dim_{D_i} \Hom_R(S_i, M), where D_i = \End_R(S_i)^{\mathrm{op}} is the opposite endomorphism ring, a division ring by Schur's lemma.^[21] Equivalently, this multiplicity equals the length of the isotypic component for S_i. In the context of composition series, the composition multiplicity [M : S_i]—the number of times S_i appears as a factor—coincides with n_i for semisimple modules, since all submodules are direct summands.^[21] Without semisimplicity, uniqueness of direct sum decompositions can fail dramatically; for instance, the free \mathbb{Z}-module of countably infinite rank admits isomorphic direct sum decompositions into rank-one summands that differ by absorbing an additional copy, such as \mathbb{Z}^{(\mathbb{N})} \cong \mathbb{Z}^{(\mathbb{N})} \oplus \mathbb{Z}.^[22] In representation theory, this uniqueness underpins the block decomposition of the module category over a semisimple algebra, where the category splits as a direct sum of blocks, each semisimple and corresponding to the multiples of a single simple module.^[23]

Endomorphism Rings

Schur's lemma

Schur's lemma asserts that for a simple left R-module S, the endomorphism ring \End_R(S) is a division ring; in other words, every nonzero R-linear map f: S \to S is an isomorphism. The proof proceeds as follows. Let f \in \End_R(S) be nonzero. Then \ker f is a proper submodule of S, so \ker f = 0 by simplicity, making f injective. Likewise, \im f is a nonzero submodule, so \im f = S, making f surjective. Hence f is bijective, and its inverse is also in \End_R(S), so every nonzero element is invertible. This result, named after Issai Schur, originated in his 1904 work on irreducible representations of finite groups over the complex numbers, where the endomorphism ring consists of scalar multiples of the identity. It has since been extended to the general setting of simple modules over associative rings. When R is a field and S is a finite-dimensional simple module, \End_R(S) is a finite-dimensional central division algebra over R. By the Frobenius theorem, over the real numbers, the possibilities include \mathbb{R}, \mathbb{C}, or the quaternions \mathbb{H}, with the latter arising, for example, in certain real representations of finite groups. As a corollary, simple modules are indecomposable: if S = A \oplus B with A and B nonzero submodules, the projection onto A parallel to B is a nontrivial idempotent in \End_R(S), but division rings admit no such idempotents other than $0

and &#36;1

Endomorphisms of semisimple modules

The endomorphism ring \operatorname{End}_R(M) of a semisimple R-module M plays a central role in understanding the structure of M, as it encodes the linear transformations that commute with the R-action. For a semisimple module M, \operatorname{End}_R(M) is itself a semisimple ring, reflecting the direct sum decomposition of M into simple summands. This ring structure arises from the vanishing of homomorphisms between non-isomorphic simple modules and the division ring nature of endomorphisms on isomorphic simples, as established by Schur's lemma. Assume M decomposes as a direct sum M = \bigoplus_{i=1}^k [S_i^{n_i}](/page/Direct_sum), where the S_i are pairwise non-isomorphic simple R-modules and each n_i \geq 1 denotes the multiplicity of S_i. Let D_i = \operatorname{End}_R(S_i), which is a division ring by Schur's lemma. Then, the endomorphism ring is isomorphic to the direct product

\operatorname{End}_R(M) \cong \prod_{i=1}^k M_{n_i}(D_i),

where M_{n_i}(D_i) denotes the ring of n_i \times n_i matrices with entries in D_i. This isomorphism follows from the computation of Hom-spaces: \operatorname{Hom}_R(S_i, S_j) = 0 for i \neq j, while endomorphisms between copies of the same S_i form the matrix ring over D_i. The direct product structure arises because endomorphisms preserve the isotypic components (direct sums of isomorphic simples) and act independently on each. As a consequence, \operatorname{End}_R(M) is a semisimple artinian ring when M is finitely generated, with its simple components corresponding to the matrix rings M_{n_i}(D_i). If M is a faithful semisimple module (meaning the annihilator of M in R is zero), then R is Morita equivalent to \operatorname{End}_R(M), implying that the category of R-modules is equivalent to the category of \operatorname{End}_R(M)-modules. The center of \operatorname{End}_R(M) is given by

Z(\operatorname{End}_R(M)) \cong \prod_{i=1}^k Z(D_i),

where Z(D_i) is the center of the division ring D_i, since the center of a matrix ring M_{n_i}(D_i) coincides with Z(D_i). A concrete example occurs in representation theory of finite groups. Consider M as a semisimple representation of a finite group G over a field k whose characteristic does not divide |G|, so all representations are semisimple by Maschke's theorem. Here, \operatorname{End}_k(M) decomposes into blocks corresponding to the isotypic components of the irreducible representations appearing in M, with each block being a full matrix algebra over the endomorphism division ring of the corresponding irreducible. This block structure mirrors the decomposition of the group algebra kG into matrix blocks over division rings, providing insight into the representation theory of G.

Semisimple Rings

Definition and examples

A ring R is defined to be left semisimple if it is semisimple as a left module over itself, denoted {}_R R.^[24] This condition implies that every left ideal of R is a direct sum of simple left submodules.^[25] Equivalently, a ring R is left semisimple if and only if it is left Artinian and its Jacobson radical J(R) is zero.^[24] In the commutative case, left Artinian is equivalent to Noetherian, so commutative semisimple rings are precisely those that are Artinian with zero Jacobson radical. Examples include finite direct products of fields, such as \mathbb{C} \times \mathbb{R}.^[1] Prominent examples of semisimple rings include full matrix rings M_n(D) over a division ring D, which decompose as direct sums of simple modules corresponding to matrix units.^[1] Another class consists of group algebras kG, where G is a finite group and k is a field whose characteristic does not divide the order of G; by Maschke's theorem, these are semisimple as they admit a complete reducibility of representations.^[3] Non-examples illustrate the boundaries of the definition. The ring of integers \mathbb{Z} is not semisimple, as it has infinite length as a module over itself due to descending chains of principal ideals like (2) \supset (4) \supset (8) \supset \cdots.^[4] Similarly, the Weyl algebra over a field of characteristic zero is simple but not semisimple, since it fails to be Artinian.^[24] A key property for commutative semisimple rings is that they are von Neumann regular, meaning for every element a \in R, there exists x \in R such that a = axa.^[26] This follows from the Artinian condition and zero Jacobson radical ensuring every principal ideal is generated by an idempotent.^[26]

Relation to modules

A semisimple Artinian ring R has the property that every left R-module is semisimple, meaning it decomposes as a direct sum of simple submodules.^[27] This follows from the Wedderburn-Artin theorem, which characterizes such rings as finite direct products of matrix rings over division rings, ensuring that all modules, including infinite ones, are semisimple, projective, and injective.^[28] Moreover, over these rings, the module category admits a complete decomposition into simples, aligning with the structural properties of semisimple modules.^[21] Conversely, a ring R is semisimple if and only if every left R-module is semisimple.^[29] This equivalence holds because free modules over a semisimple R (as a module over itself) are direct sums of copies of R, hence semisimple, and every module is a quotient of a free module, with quotients of semisimple modules remaining semisimple.^[29] By the Wedderburn-Artin theorem, this occurs precisely when R is a finite direct product of full matrix rings over division rings.^[28] Semisimple rings are Morita equivalent to finite products of division rings, as each matrix ring over a division ring D is Morita equivalent to D itself via the bimodule of row vectors.^[30] Under this equivalence, left modules over the semisimple ring correspond to vector spaces over the product of division rings, preserving the semisimple structure and facilitating the study of module categories.^[30] In representation theory, semisimple algebras provide a setting where every finite-dimensional representation is completely reducible, decomposing directly into a sum of irreducible representations without extensions.^[31] This property is fundamental for classifying representations of groups or Lie algebras over semisimple coefficient rings.^[31]

Advanced Aspects

Artinian semisimple rings

A ring R is left Artinian if it satisfies the descending chain condition on left ideals, meaning that every descending chain of left ideals stabilizes after finitely many steps.^[32] Equivalently, every nonempty set of left ideals has a minimal element.^[33] Left Artinian rings are also right Artinian and Noetherian, with the regular module having finite length.^[28] The Wedderburn–Artin theorem provides the complete structure of semisimple Artinian rings. It states that a left Artinian semisimple ring R is isomorphic to a finite direct product R \cong \prod_{i=1}^r M_{n_i}(D_i), where each D_i is a division ring and each n_i is a positive integer; the decomposition is unique up to permutation of the factors and isomorphism of the D_i.^[32] This result, originally due to Wedderburn for simple algebras and extended by Artin, classifies such rings as finite matrix rings over division rings. The proof of the Wedderburn–Artin theorem proceeds by first showing that a semisimple Artinian ring is a finite direct sum of minimal left ideals, using the Artinian condition to ensure finiteness. For the simple components, the double centralizer theorem is applied: in a finite-dimensional central simple algebra A over a field with simple subalgebra B, the centralizer C_A(B) is simple, B = C_A(C_A(B)), and \dim B \cdot \dim C_A(B) = \dim A. Jacobson's density theorem further establishes that for a primitive ideal, the action of the ring is dense in the endomorphism ring of a faithful simple module, leading to the matrix ring structure over a division ring.^[32] Semisimple Artinian rings have global dimension zero, implying that every left (or right) module is projective and injective.^[28] If the ring is commutative, it decomposes as a finite direct product of fields.^[32] Modern extensions consider graded or twisted versions, such as groupoid-graded semisimple rings, which may involve infinite products of Artinian components under certain grading conditions.^[34]

Jacobson semisimplification

The Jacobson radical J(R) of a ring R is defined as the intersection of all maximal left ideals of R.^[35] This ideal captures the "non-semisimple" part of the ring in the sense that R is semisimple if and only if J(R) = 0.^[1] The Jacobson semisimplification of R is the quotient ring R / J(R), which is always semiprimitive (i.e., has zero Jacobson radical) and thus Jacobson semisimple.^[36] A ring R is semisimple if and only if it is semisimple as a module over itself and J(R) = 0, meaning the regular module {}_R R decomposes as a direct sum of simple submodules.^[1] In non-Artinian rings, semisimple modules exist abundantly—for instance, any direct sum of simple modules is semisimple—but the ring itself may fail to be semisimple even if J(R) = 0. A classic example is the infinite direct product of copies of a field k, denoted \prod_{i \in I} k for infinite index set I; here J(R) = 0, yet R as a left module over itself is a direct product rather than a direct sum of simples, so it is not semisimple.^[37] Artinian semisimple rings form a special case where J(R) = 0 implies a finite direct sum decomposition into simple Artinian rings.^[1] Applications of Jacobson semisimplification appear in algebraic geometry, particularly in deformation theory, where the semisimple quotient of a deformation ring encodes the semisimple part of Galois representations or algebraic structures under infinitesimal deformations. For example, in the study of potentially semi-stable pseudodeformation rings for Galois representations, the quotient by the Jacobson radical yields a semisimple ring that classifies the semisimple types compatible with the deformation functor.^[38] Non-commutative examples highlight the distinction: while finite matrix rings over division rings are semisimple Artinian with J(R) = 0, the Weyl algebra over a field provides a non-Artinian simple ring with J(R) = 0 that is not semisimple as a module over itself, as its regular module does not decompose into simples.^[2]

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[PDF] §1. Semisimple rings - Penn Math
(1.1) Definition A ring R with 1 is semisimple, or left semisimple to be precise, if the free left R-module underlying R is a sum of simple R-module. (1.2) ...
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[PDF] 3. Semisimple rings
Definition 3.4. A ring R is semisimple if it semisimple as a left module over itself. A ring R is simple if it is semisimple and if it has exactly one type of ...
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von Neumann regular ring in nLab
Jan 6, 2025 · A von Neumann regular ring or absolutely flat ring (Lombardi & Quitté (2010)) is one where every principal left ideal or principal right ideal is generated by ...Idea · Definition · Commutative von Neumann... · Non-algebraicity of von...
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[PDF] Wedderburn-Artin theorem in Group representations
Mar 22, 2021 · Theorem. If R is a semisimple artinian ring, then every left (right) R-module is a direct sum of simple left (right) R-modules. Corollary. If ...
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[PDF] facts on semi-simple artinian rings
For all i ≥ 1, Ji/Ji+1 is a finite direct sum of simple modules and in particular has finite length. proof: Since Ji ⊆ R, then Ji is an artinian R-module and ...
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[PDF] 9. Simple and semisimple rings
A semisimple ring satisfying any of the equivalent conditions below is called a simple ring. The equivalence of the conditions is easy to see given the ...
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[PDF] Morita Theory for Rings and Semigroups
The notion of Morita equivalence for rings defines a relationship between rings in terms of their module categories being equivalent in the sense of category ...
[32]
[PDF] 25. Representation theory of semisimple Lie algebras
Recall that by Theorem 18.9, every finite dimensional representation of g is completely reducible, so to classify finite dimensional representations it suffices ...
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Advanced Algebra - Project Euclid
The hypothesis. Page 15. 88. II. Wedderburn-Artin Ring Theory left Artinian may therefore be regarded as a useful generalization of finite dimen- sionality.<|control11|><|separator|>
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[PDF] Advanced Algebra
The hypothesis. Page 14. 88. XI. Wedderburn-Artin Ring Theory left Artinian may therefore be regarded as a useful generalization of finite dimen- sionality.Missing: book | Show results with:book
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Groupoid Graded Semisimple Rings - ScienceDirect.com
Sep 3, 2025 · Semisimple rings and the Wedderburn-Artin Theorem lie at the basis of classical Ring Theory. Later, the study of group graded structures ...Missing: reliable | Show results with:reliable<|control11|><|separator|>
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[PDF] The Jacobson radical
Apr 1, 2015 · For example, J(Z) = 0 but Z is not semisimple. The condition J(R) = 0 is sometimes called “Jacobson semisimple”. Now consider the quotient ...
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[PDF] Section IX.2. The Jacobson Radical
Oct 6, 2018 · If R is a ring, then the quotient ring R/J(R) is semisimple. Lemma IX.2.15. Let R be a ring and a ∈ R. (i) If −a2 is left quasi-regular ...
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Chapter 32, Simple and Semisimple Rings and Modules
It's interesting to see why an infinite direct product of simple modules does not build a semisimple ring. Let R be the direct product of infinitely many ...
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Algebraic families of Galois representations and potentially semi ...
We remark that the ring-theoretic properties of the potentially semi-stable pseudodeformation rings in Theorem C are deduced from the geometric properties of ...<|separator|>