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Semisimple module
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In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups over fields of characteristic zero, are semisimple rings. An Artinian ring is initially understood via its largest semisimple quotient. The structure of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings.

For a group-theory analog of the same notion, see Semisimple representation.

Definition

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A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules.

For a module M, the following are equivalent:

  1. M is semisimple; i.e., a direct sum of irreducible modules.
  2. M is the sum of its irreducible submodules.
  3. Every submodule of M is a direct summand: for every submodule N of M, there is a complement P such that M = N ⊕ P.

For the proof of the equivalences, see Semisimple representation § Equivalent characterizations.

The most basic example of a semisimple module is a module over a field, i.e., a vector space. On the other hand, the ring Z of integers is not a semisimple module over itself, since the submodule 2Z is not a direct summand.

Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules.

Let A be an algebra over a field K. Then a left module M over A is said to be absolutely semisimple if, for any field extension F of K, F ⊗K M is a semisimple module over F ⊗K A.

Properties

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  • If M is semisimple and N is a submodule, then N and M / N are also semisimple.
  • An arbitrary direct sum of semisimple modules is semisimple.
  • A module M is finitely generated and semisimple if and only if it is Artinian and its radical is zero.

Endomorphism rings

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Semisimple rings

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A ring is said to be (left-)semisimple if it is semisimple as a left module over itself.[2] Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary, and one can speak of semisimple rings without ambiguity.

A semisimple ring may be characterized in terms of homological algebra: namely, a ring R is semisimple if and only if any short exact sequence of left (or right) R-modules splits. That is, for a short exact sequence

there exists s : C → B such that the composition g ∘ s : C → C is the identity. The map s is known as a section. From this it follows that

or in more exact terms

In particular, any module over a semisimple ring is injective and projective. Since "projective" implies "flat", a semisimple ring is a von Neumann regular ring.

Semisimple rings are of particular interest to algebraists. For example, if the base ring R is semisimple, then all R-modules would automatically be semisimple. Furthermore, every simple (left) R-module is isomorphic to a minimal left ideal of R, that is, R is a left Kasch ring.

Semisimple rings are both Artinian and Noetherian. From the above properties, a ring is semisimple if and only if it is Artinian and its Jacobson radical is zero.

If an Artinian semisimple ring contains a field as a central subring, it is called a semisimple algebra.

Examples

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Simple rings

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Main article: Simple ring

One should beware that despite the terminology, not all simple rings are semisimple. The problem is that the ring may be "too big", that is, not (left/right) Artinian. In fact, if R is a simple ring with a minimal left/right ideal, then R is semisimple.

Classic examples of simple, but not semisimple, rings are the Weyl algebras, such as the Q-algebra

which is a simple noncommutative domain. These and many other nice examples are discussed in more detail in several noncommutative ring theory texts, including chapter 3 of Lam's text, in which they are described as nonartinian simple rings. The module theory for the Weyl algebras is well studied and differs significantly from that of semisimple rings.

Jacobson semisimple

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Main article: Semiprimitive ring

A ring is called Jacobson semisimple (or J-semisimple or semiprimitive) if the intersection of the maximal left ideals is zero, that is, if the Jacobson radical is zero. Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself. A J-semisimple ring is semisimple if and only if it is an artinian ring, so semisimple rings are often called artinian semisimple rings to avoid confusion.

For example, the ring of integers, Z, is J-semisimple, but not artinian semisimple.

See also

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Citations

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  1. ^ Lam 2001, p. 62
  2. ^ Sengupta 2012, p. 125
  3. ^ Bourbaki 2012, p. 133, VIII

References

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Semisimple module

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In , a semisimple module over a ring RR is an RR-module that decomposes as a of simple RR-modules, where simple modules are nonzero modules with no proper nonzero submodules. This decomposition may be finite or infinite, though in many contexts—such as finitely generated modules—it is finite. Semisimplicity admits several equivalent characterizations: a module MM is semisimple if and only if it is a sum (not necessarily direct) of simple submodules; if every submodule of MM is a direct summand; or if every surjective homomorphism from MM onto another module splits. These properties highlight the "reducibility" of semisimple modules, meaning they break down completely into irreducible building blocks without further complications from extensions. Key properties of semisimple modules include the uniqueness of their simple summands up to 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. Finite-length semisimple modules are both Noetherian and Artinian, ensuring well-behaved ascending and descending chains of submodules. In , 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. 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.

Fundamental Concepts

Simple modules

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

Direct sums and summands

The direct sum of a family of modules {Mi}i∈I\{M_i\}_{i \in I} over a ring RR, also known as the external direct sum and denoted ⨁i∈IMi\bigoplus_{i \in I} M_i, consists of all families (mi)i∈I(m_i)_{i \in I} with mi∈Mim_i \in M_i and only finitely many mim_i nonzero, equipped with componentwise addition and scalar multiplication. This construction ensures that each MiM_i embeds naturally as a submodule via the map sending m∈Mim \in M_i to the family with mm in the ii-th position and zeros elsewhere, and the sum of these images is direct. An internal direct sum arises when a module MM is expressed as the sum of submodules Nj⊆MN_j \subseteq M (j∈Jj \in J) such that every element of MM is a finite sum of elements from the NjN_j and the intersection of any NkN_k with the sum of the others is zero. In this case, M≅⨁j∈JNjM \cong \bigoplus_{j \in J} N_j as RR-modules, with the isomorphism identifying each element uniquely as such a finite sum. A submodule N⊆MN \subseteq M is a direct summand if there exists another submodule K⊆MK \subseteq M such that MM is the internal direct sum of NN and KK. Direct sums of modules preserve exactness of sequences in the category of RR-modules, meaning that if $0 \to A_i \to B_i \to C_i \to 0isexactforeachis exact for eachi, then $0 \to \bigoplus_i A_i \to \bigoplus_i B_i \to \bigoplus_i C_i \to 0 is also exact. This property holds because direct sums are both products and coproducts (biproducts) in this abelian category, facilitating homological computations. Idempotents in the endomorphism ring End⁡R(M)\operatorname{End}_R(M) of a module MM correspond precisely to direct sum decompositions of MM. Specifically, if e∈End⁡R(M)e \in \operatorname{End}_R(M) satisfies e2=ee^2 = e, then M=im⁡(e)⊕ker⁡(e)M = \operatorname{im}(e) \oplus \ker(e), where im⁡(e)\operatorname{im}(e) is the image of ee and ker⁡(e)\ker(e) is its kernel. Conversely, given a decomposition M=N⊕KM = N \oplus K, the projection onto NN along KK defines an idempotent endomorphism with image NN and kernel KK. 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.

Definition and Characterizations

Primary definition

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

Equivalent conditions

A module MM over a ring RR is semisimple if and only if every submodule of MM is a direct summand, that is, for every submodule N⊆MN \subseteq M, there exists a submodule N′⊆MN' \subseteq M such that M=N⊕N′M = N \oplus N'. This condition is equivalent to MM being a direct sum of simple submodules and also to MM being a (not necessarily direct) sum of simple submodules. These equivalences are standard and can be found, with proofs, in Lang's Algebra, Chapter XVII, §2. To prove the equivalence of the three conditions, denote them as:
  1. M=∑EiM = \sum E_i for some simple submodules EiE_i.
  2. M=⨁EjM = \bigoplus E_j for some simple submodules EjE_j.
  3. Every submodule of MM is a direct summand.
(1) ⇒ (2): Suppose M=∑iEiM = \sum_{i} E_i with each EiE_i simple. By Zorn's lemma, there exists a maximal subfamily {Ej}\{E_j\} such that E′=⨁EjE' = \bigoplus E_j is direct. Then E′=ME' = M: for any EkE_k, the intersection Ek∩E′E_k \cap E' is a submodule of EkE_k, hence either {0}\{0\} or EkE_k. If {0}\{0\}, adjoining EkE_k would yield a larger direct sum, contradicting maximality. Thus Ek⊆E′E_k \subseteq E', so M=E′M = E'. Zorn's lemma is unnecessary when MM has finite length or is finite-dimensional, as maximality follows from finiteness arguments. (2) ⇒ (3): Suppose M=⨁EjM = \bigoplus E_j with each EjE_j simple. For any submodule F⊆MF \subseteq M, consider a maximal direct sum E′=⨁EjE' = \bigoplus E_j of simple submodules with E′∩F={0}E' \cap F = \{0\}. By a similar maximality argument applied to simple submodules disjoint from FF, M=E′⊕FM = E' \oplus F. (3) ⇒ (1): Let \soc(M)\soc(M) be the sum of all simple submodules of MM. By (3), M=\soc(M)⊕KM = \soc(M) \oplus K for some submodule KK. Suppose K≠{0}K \neq \{0\}. Take 0≠v∈K0 \neq v \in K. Then Rv≠{0}Rv \neq \{0\} is contained in KK. There exists a maximal proper submodule LL of RvRv (using Zorn's lemma if necessary). By (3), Rv=L⊕SRv = L \oplus S for some S≠{0}S \neq \{0\}. Since LL is maximal proper in RvRv, the quotient Rv/LRv/L is simple, and S≅Rv/LS \cong Rv/L is simple. But S⊆Rv⊆KS \subseteq Rv \subseteq K, so KK contains a nonzero simple submodule, contradicting that all simple submodules are contained in \soc(M)\soc(M) (since K∩\soc(M)={0}K \cap \soc(M) = \{0\}). Thus K={0}K = \{0\}, so M=\soc(M)M = \soc(M), the sum of simple submodules. Again, Zorn's lemma can be avoided when MM has finite length, as submodules of finite length modules satisfy the ascending chain condition. For modules of finite length, semisimplicity is equivalent to the existence of a with simple factors in which every short arising from consecutive terms splits. In such cases, the Jordan-Hölder ensures all have the same and isomorphic factors up to permutation, and the splitting property follows from the decomposition into simples, as each step in the series corresponds to a direct summand. More generally, MM is semisimple if and only if every short 0→N→M→Q→00 \to N \to M \to Q \to 0 with NN and QQ semisimple splits, since submodules and quotients of semisimple modules inherit semisimplicity, allowing inductive . Another characterization is that the socle \soc(M)\soc(M), the sum of all simple submodules of MM, equals MM. This holds because the socle is the largest semisimple submodule, and if it coincides with MM, then MM is semisimple by the sum-of-simples condition. Dually, for finitely generated modules over finite-dimensional algebras, semisimplicity is equivalent to the radical \rad(M)=0\rad(M) = 0, where the radical is the intersection of all maximal submodules (or kernels of epimorphisms onto simples). 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. In certain ring classes, such as semisimple artinian rings, a module MM is semisimple if and only if it is both projective and injective. Over such rings, all modules are semisimple, hence both projective (as direct summands of frees) and injective (via splitting of monomorphisms). However, this equivalence fails in general; for instance, over the integers Z\mathbb{Z}, the module Z/pZ\mathbb{Z}/p\mathbb{Z} (for prime pp) is simple, hence semisimple, but neither projective (as projectives over Z\mathbb{Z} are free) nor injective (as injectives are divisible groups). Over commutative rings, semisimplicity coincides with the module being completely reducible, meaning every submodule is complemented by a direct summand. 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. The submodule summand property ensures no proper essential submodules exist, as any essential N⊊MN \subsetneq M would intersect its complement trivially, violating essentiality.

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 RR-module MM is isomorphic to a direct sum of simple submodules. To prove this, consider the set of all families of simple submodules whose direct sum is a submodule of MM; this set is nonempty since it contains the empty family and single simple submodules. By Zorn's lemma, there exists a maximal such family {Sα}α∈I\{S_\alpha\}_{\alpha \in I}. If the direct sum ⨁α∈ISα≠M\bigoplus_{\alpha \in I} S_\alpha \neq M, then there is a simple submodule S⊆MS \subseteq M not contained in this sum, and since MM is semisimple, SS complements the sum as a direct summand, contradicting maximality. Thus, M≅⨁α∈ISαM \cong \bigoplus_{\alpha \in I} S_\alpha. An alternative proof uses transfinite induction on the partial order of direct sums of simples, extending maximal decompositions at limit ordinals. Decompositions of semisimple modules may be finite or infinite. If MM is artinian and semisimple, then its decomposition into simple summands is finite. 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 Z\mathbb{Z} of integers, the infinite direct sum ⨁n=1∞Z/pZ\bigoplus_{n=1}^\infty \mathbb{Z}/p\mathbb{Z} (for a fixed prime pp) is semisimple, as each Z/pZ\mathbb{Z}/p\mathbb{Z} is a simple Z\mathbb{Z}-module and direct sums preserve semisimplicity, but it is not artinian. The simple summands in a decomposition of a semisimple module MM can be grouped by isomorphism classes into isotypic components. For each isomorphism class of simple modules represented by VV, the VV-isotypic component of MM is the direct sum of all submodules of MM isomorphic to VV, which takes the form V⊗kkVV \otimes_k k_V, where kVk_V is a vector space over the division ring End⁡R(V)op\operatorname{End}_R(V)^{\mathrm{op}} with dimension equal to the multiplicity of VV in MM, and MM is the direct sum over all such distinct simples VV of these components. For semisimple modules M=⨁iSiniM = \bigoplus_i S_i^{n_i} and N=⨁jTjmjN = \bigoplus_j T_j^{m_j} (with Si,TjS_i, T_j simple), the spaces Hom⁡R(M,N)\operatorname{Hom}_R(M, N) and Ext⁡R1(M,N)\operatorname{Ext}^1_R(M, N) decompose according to the simple components, as both functors respect direct sums: Hom⁡R(M,N)≅⨁i,jHom⁡R(Sini,Tjmj)\operatorname{Hom}_R(M, N) \cong \bigoplus_{i,j} \operatorname{Hom}_R(S_i^{n_i}, T_j^{m_j}) and similarly for Ext⁡R1(M,N)\operatorname{Ext}^1_R(M, N), with nonzero terms only when Si≅TjS_i \cong T_j. In particular, for the endomorphism ring, if M=⨁iSiniM = \bigoplus_i S_i^{n_i} with each End⁡R(Si)\operatorname{End}_R(S_i) a DiD_i, then dim⁡Hom⁡R(M,M)=∑ini2dim⁡Di\dim \operatorname{Hom}_R(M, M) = \sum_i n_i^2 \dim D_i. End⁡R(M)≅∏iMni(Di),dim⁡End⁡R(M)=∑ini2dim⁡Di\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 MM of finite length over a ring RR, the Krull–Schmidt–Azumaya theorem guarantees that any direct sum decomposition M≅⨁iSiniM \cong \bigoplus_i S_i^{n_i} into simple summands SiS_i (up to isomorphism) is unique up to isomorphism of the summands and permutation of the indices. This uniqueness extends to general semisimple modules through their isotypic decomposition, where the isotypic component corresponding to a simple module SS—the direct sum of all summands isomorphic to SS—is uniquely determined as the maximal submodule on which the endomorphism ring acts via the division ring \EndR(S)\End_R(S). The multiplicity nin_i of a simple summand SiS_i in the decomposition is an invariant given by ni=dim⁡Di\HomR(Si,M)n_i = \dim_{D_i} \Hom_R(S_i, M), where Di=\EndR(Si)opD_i = \End_R(S_i)^{\mathrm{op}} is the opposite endomorphism ring, a division ring by Schur's lemma. Equivalently, this multiplicity equals the length of the isotypic component for SiS_i. In the context of composition series, the composition multiplicity [M:Si][M : S_i]—the number of times SiS_i appears as a factor—coincides with nin_i for semisimple modules, since all submodules are direct summands. Without , uniqueness of decompositions can fail dramatically; for instance, the free Z\mathbb{Z}-module of countably infinite rank admits isomorphic decompositions into rank-one summands that differ by absorbing an additional copy, such as Z(N)≅Z(N)⊕Z\mathbb{Z}^{(\mathbb{N})} \cong \mathbb{Z}^{(\mathbb{N})} \oplus \mathbb{Z}. In , this uniqueness underpins the block decomposition of the module category over a semisimple , where the category splits as a of blocks, each semisimple and corresponding to the multiples of a single simple module.

Endomorphism Rings

Schur's lemma

Schur's lemma asserts that for a simple left RR-module SS, the ring \EndR(S)\End_R(S) is a ; in other words, every nonzero RR- f:S→Sf: S \to S is an . The proof proceeds as follows. Let f∈\EndR(S)f \in \End_R(S) be nonzero. Then ker⁡f\ker f is a proper submodule of SS, so ker⁡f=0\ker f = 0 by simplicity, making ff injective. Likewise, \imf\im f is a nonzero submodule, so \imf=S\im f = S, making ff surjective. Hence ff is bijective, and its inverse is also in \EndR(S)\End_R(S), so every nonzero element is invertible. This result, named after , originated in his 1904 work on irreducible representations of finite groups over the complex numbers, where the ring consists of scalar multiples of the identity. It has since been extended to the general setting of simple modules over associative rings. When RR is a field and SS is a finite-dimensional simple module, \EndR(S)\End_R(S) is a finite-dimensional central over RR. By the Frobenius theorem, over the real numbers, the possibilities include R\mathbb{R}, C\mathbb{C}, or the quaternions H\mathbb{H}, with the latter arising, for example, in certain real representations of finite groups. As a , simple modules are indecomposable: if S=A⊕BS = A \oplus B with AA and BB nonzero submodules, the projection onto AA parallel to BB is a nontrivial idempotent in \EndR(S)\End_R(S), but division rings admit no such idempotents other than $0 and $1.

Endomorphisms of semisimple modules

The endomorphism ring End⁡R(M)\operatorname{End}_R(M) of a semisimple RR-module MM plays a central role in understanding the structure of MM, as it encodes the linear transformations that commute with the RR-action. For a semisimple module MM, End⁡R(M)\operatorname{End}_R(M) is itself a semisimple ring, reflecting the direct sum decomposition of MM 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 MM decomposes as a direct sum M=⨁i=1k[Sini](/page/Directsum)M = \bigoplus_{i=1}^k [S_i^{n_i}](/page/Direct_sum), where the SiS_i are pairwise non-isomorphic simple RR-modules and each ni≥1n_i \geq 1 denotes the multiplicity of SiS_i. Let Di=End⁡R(Si)D_i = \operatorname{End}_R(S_i), which is a by . Then, the ring is isomorphic to the End⁡R(M)≅∏i=1kMni(Di),\operatorname{End}_R(M) \cong \prod_{i=1}^k M_{n_i}(D_i), where Mni(Di)M_{n_i}(D_i) denotes the ring of ni×nin_i \times n_i matrices with entries in DiD_i. This isomorphism follows from the of Hom-spaces: Hom⁡R(Si,Sj)=0\operatorname{Hom}_R(S_i, S_j) = 0 for i≠ji \neq j, while endomorphisms between copies of the same SiS_i form the matrix ring over DiD_i. The structure arises because endomorphisms preserve the isotypic components (direct sums of isomorphic simples) and act independently on each. As a consequence, End⁡R(M)\operatorname{End}_R(M) is a semisimple artinian ring when MM is finitely generated, with its simple components corresponding to the matrix rings Mni(Di)M_{n_i}(D_i). If MM is a faithful semisimple module (meaning the annihilator of MM in RR is zero), then RR is Morita equivalent to End⁡R(M)\operatorname{End}_R(M), implying that the category of RR-modules is equivalent to the category of End⁡R(M)\operatorname{End}_R(M)-modules. The center of End⁡R(M)\operatorname{End}_R(M) is given by Z(End⁡R(M))≅∏i=1kZ(Di),Z(\operatorname{End}_R(M)) \cong \prod_{i=1}^k Z(D_i), where Z(Di)Z(D_i) is the center of the division ring DiD_i, since the center of a matrix ring Mni(Di)M_{n_i}(D_i) coincides with Z(Di)Z(D_i). A concrete example occurs in representation theory of finite groups. Consider MM as a semisimple representation of a finite group GG over a field kk whose characteristic does not divide ∣G∣|G|, so all representations are semisimple by Maschke's theorem. Here, End⁡k(M)\operatorname{End}_k(M) decomposes into blocks corresponding to the isotypic components of the irreducible representations appearing in MM, 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 kGkG into matrix blocks over division rings, providing insight into the representation theory of GG.

Semisimple Rings

Definition and examples

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

Relation to modules

A semisimple Artinian ring RR has the property that every left RR-module is semisimple, meaning it decomposes as a direct sum of simple submodules. 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. Moreover, over these rings, the module category admits a complete decomposition into simples, aligning with the structural properties of semisimple modules. Conversely, a ring RR is semisimple if and only if every left RR-module is semisimple. This equivalence holds because free modules over a semisimple RR (as a module over itself) are direct sums of copies of RR, hence semisimple, and every module is a quotient of a free module, with quotients of semisimple modules remaining semisimple. By the Wedderburn-Artin theorem, this occurs precisely when RR is a finite direct product of full matrix rings over division rings. Semisimple rings are Morita equivalent to finite products of division rings, as each matrix ring over a division ring DD is Morita equivalent to DD itself via the bimodule of row vectors. 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. 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. This property is fundamental for classifying representations of groups or Lie algebras over semisimple coefficient rings.

Advanced Aspects

Artinian semisimple rings

A ring RR 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. Equivalently, every nonempty set of left ideals has a minimal element. Left Artinian rings are also right Artinian and Noetherian, with the regular module having finite length. The Wedderburn–Artin theorem provides the complete structure of semisimple Artinian rings. It states that a left Artinian semisimple ring RR is isomorphic to a finite direct product R≅∏i=1rMni(Di)R \cong \prod_{i=1}^r M_{n_i}(D_i), where each DiD_i is a division ring and each nin_i is a positive integer; the decomposition is unique up to permutation of the factors and isomorphism of the DiD_i. 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 AA over a field with simple subalgebra BB, the centralizer CA(B)C_A(B) is simple, B=CA(CA(B))B = C_A(C_A(B)), and dim⁡B⋅dim⁡CA(B)=dim⁡A\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. Semisimple Artinian rings have global dimension zero, implying that every left (or right) module is projective and injective. If the ring is commutative, it decomposes as a finite of fields. 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.

Jacobson semisimplification

The Jacobson radical J(R)J(R) of a ring RR is defined as the intersection of all maximal left ideals of RR. This ideal captures the "non-semisimple" part of the ring in the sense that RR is semisimple if and only if J(R)=0J(R) = 0. The Jacobson semisimplification of RR is the quotient ring R/J(R)R / J(R), which is always semiprimitive (i.e., has zero Jacobson radical) and thus Jacobson semisimple. A ring RR is semisimple if and only if it is semisimple as a module over itself and J(R)=0J(R) = 0, meaning the regular module RR{}_R R decomposes as a direct sum of simple submodules. 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)=0J(R) = 0. A classic example is the infinite direct product of copies of a field kk, denoted ∏i∈Ik\prod_{i \in I} k for infinite index set II; here J(R)=0J(R) = 0, yet RR as a left module over itself is a direct product rather than a direct sum of simples, so it is not semisimple. Artinian semisimple rings form a special case where J(R)=0J(R) = 0 implies a finite direct sum decomposition into simple Artinian rings. 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. Non-commutative examples highlight the distinction: while finite matrix rings over division rings are semisimple Artinian with J(R)=0J(R) = 0, the Weyl algebra over a field provides a non-Artinian simple ring with J(R)=0J(R) = 0 that is not semisimple as a module over itself, as its regular module does not decompose into simples.

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