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In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory.

In this article, all modules will be assumed to be right unital modules over a ring R.

Examples

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Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups. These are the cyclic groups of prime order.

If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal. Conversely, if I is not minimal, then there is a non-zero right ideal J properly contained in I. J is a right submodule of I, so I is not simple.

If I is a right ideal of R, then the quotient module R/I is simple if and only if I is a maximal right ideal: If M is a non-zero proper submodule of R/I, then the preimage of M under the quotient map R → R/I is a right ideal which is not equal to R and which properly contains I. Therefore, I is not maximal. Conversely, if I is not maximal, then there is a right ideal J properly containing I. The quotient map R/I → R/J has a non-zero kernel which is not equal to R/I, and therefore R/I is not simple.

Every simple R-module is isomorphic to a quotient R/m where m is a maximal right ideal of R.[1] By the above paragraph, any quotient R/m is a simple module. Conversely, suppose that M is a simple R-module. Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x. The statement that xR = M is equivalent to the surjectivity of the homomorphism R → M that sends r to xr. The kernel of this homomorphism is a right ideal I of R, and a standard theorem states that M is isomorphic to R/I. By the above paragraph, we find that I is a maximal right ideal. Therefore, M is isomorphic to a quotient of R by a maximal right ideal.

If k is a field and G is a group, then a group representation of G is a left module over the group ring k[G] (for details, see the main page on this relationship).[2] The simple k[G]-modules are also known as irreducible representations. A major aim of representation theory is to understand the irreducible representations of groups.

Basic properties of simple modules

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The simple modules are precisely the modules of length 1; this is a reformulation of the definition.

Every simple module is indecomposable, but the converse is in general not true.

Every simple module is cyclic, that is it is generated by one element.

Not every module has a simple submodule; consider for instance the Z-module Z in light of the first example above.

Let M and N be (left or right) modules over the same ring, and let f : M → N be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective because the kernel of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image of f is a submodule of N. If M = N, then f is an endomorphism of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism. Consequently, the endomorphism ring of any simple module is a division ring. This result is known as Schur's lemma.

The converse of Schur's lemma is not true in general. For example, the Z-module Q is not simple, but its endomorphism ring is isomorphic to the field Q.

Simple modules and composition series

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Main article: Composition series

If M is a module which has a non-zero proper submodule N, then there is a short exact sequence

A common approach to proving a fact about M is to show that the fact is true for the center term of a short exact sequence when it is true for the left and right terms, then to prove the fact for N and M/N. If N has a non-zero proper submodule, then this process can be repeated. This produces a chain of submodules

In order to prove the fact this way, one needs conditions on this sequence and on the modules Mi /Mi+1. One particularly useful condition is that the length of the sequence is finite and each quotient module Mi /Mi+1 is simple. In this case the sequence is called a composition series for M. In order to prove a statement inductively using composition series, the statement is first proved for simple modules, which form the base case of the induction, and then the statement is proved to remain true under an extension of a module by a simple module. For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull–Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.

The Jordan–Hölder theorem and the Schreier refinement theorem describe the relationships amongst all composition series of a single module. The Grothendieck group ignores the order in a composition series and views every finite length module as a formal sum of simple modules. Over semisimple rings, this is no loss as every module is a semisimple module and so a direct sum of simple modules. Ordinary character theory provides better arithmetic control, and uses simple CG modules to understand the structure of finite groups G. Modular representation theory uses Brauer characters to view modules as formal sums of simple modules, but is also interested in how those simple modules are joined together within composition series. This is formalized by studying the Ext functor and describing the module category in various ways including quivers (whose nodes are the simple modules and whose edges are composition series of non-semisimple modules of length 2) and Auslander–Reiten theory where the associated graph has a vertex for every indecomposable module.

The Jacobson density theorem

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Main article: Jacobson density theorem

An important advance in the theory of simple modules was the Jacobson density theorem. The Jacobson density theorem states:

Let U be a simple right R-module and let D = EndR(U). Let A be any D-linear operator on U and let X be a finite D-linearly independent subset of U. Then there exists an element r of R such that x⋅A = x⋅r for all x in X.[3]

In particular, any primitive ring may be viewed as (that is, isomorphic to) a ring of D-linear operators on some D-space.

A consequence of the Jacobson density theorem is Wedderburn's theorem; namely that any right Artinian simple ring is isomorphic to a full matrix ring of n-by-n matrices over a division ring for some n. This can also be established as a corollary of the Artin–Wedderburn theorem.

See also

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References

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Simple module

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In module theory, a simple module (also known as an irreducible module) over a ring RR is defined as a nonzero RR-module MM that possesses no proper nontrivial submodules, meaning the only submodules of MM are {0}\{0\} and MM itself.[1][2][3] This structure captures the most indecomposable units in the category of modules, analogous to simple groups in group theory, and serves as a foundational concept for understanding module decompositions and ring properties.[1][3] Simple modules exhibit several key properties that highlight their atomic nature. Every simple module is cyclic, generated by any single nonzero element, and can be expressed as isomorphic to R/IR/I, where II is a maximal left ideal of RR.[2][3] By Schur's lemma, the endomorphism ring EndR(M)\mathrm{End}_R(M) of a simple module MM is a division ring (or skew field), ensuring that any nonzero endomorphism is invertible.[1][2][3] Furthermore, any nonzero homomorphism between simple modules is an isomorphism, implying that non-isomorphic simple modules have zero Hom-spaces between them.[1][2] These modules play a central role in broader module theory, particularly as the building blocks of semisimple modules, which decompose as direct sums of simple modules.[3] Every nonzero ring admits at least one simple module, often constructed as a quotient by a maximal submodule.[1] Classic examples include the cyclic group Z/pZ\mathbb{Z}/p\mathbb{Z} as a simple Z\mathbb{Z}-module for a prime pp, or one-dimensional vector spaces over a field viewed as modules over that field.[2][3] In representation theory and Artinian rings, simple modules underpin composition series and the structure of finite-length modules, with their injective hulls often providing insights into more complex extensions.[1][3]

Fundamentals

Definition

In module theory, a simple module over a ring $ R $ is defined as a nonzero left $ R $-module $ M $ (or right $ R $-module, depending on the convention) such that the only submodules of $ M $ are the zero submodule $ {0} $ and $ M $ itself.[2][3] This condition emphasizes that simplicity arises from the ring action on the module, where submodules are subsets closed under addition and scalar multiplication by elements of $ R $. In the category $ R $-Mod of left $ R $-modules, simple modules serve as the basic indecomposable building blocks analogous to atoms in lattice theory.[3] Equivalent characterizations of simple modules include the following: $ M $ is simple if and only if it is nonzero and every nonzero submodule of $ M $ equals $ M $; or, equivalently, every nonzero element of $ M $ generates $ M $ as an $ R $-module, meaning $ M $ is cyclic with no proper nonzero submodules.[2][3] Another formulation states that $ M $ is simple if and only if it is isomorphic to $ R / I $ for some maximal left ideal $ I $ of $ R $.[3] These equivalences highlight the module's minimal structure under the ring's action. The concept of simple modules parallels that of simple groups in group theory and emerged in the early 20th century with the development of abstract algebra and module theory.

Basic properties

A simple module over a ring RR has the property that any nonzero RR-module homomorphism between two simple modules is necessarily an isomorphism.[4] Consequently, for simple modules SS and TT, the Hom space \HomR(S,T)\Hom_R(S, T) is either zero, in which case there are no nonzero homomorphisms, or its nonzero elements are all isomorphisms.[2] The endomorphism ring \EndR(M)\End_R(M) of a simple module MM is a division ring, as established by Schur's lemma: every nonzero endomorphism of MM is invertible.[4] This follows from the fact that the kernel of a nonzero endomorphism would be a proper submodule, contradicting the simplicity of MM.[2] Every simple module is cyclic, meaning it is generated by a single element; moreover, it is generated by any nonzero element, since the submodule generated by such an element cannot be proper.[4] Simple modules are both Noetherian and Artinian, owing to their composition length being exactly 1, which implies the absence of infinite ascending or descending chains of submodules.[5] For simple modules SS and TT, the Hom space \HomR(S,T)\Hom_R(S, T), viewed as a module over the division ring k=\EndR(S)opk = \End_R(S)^{\mathrm{op}}, has dimension at most 1; it is 1-dimensional precisely when S≅TS \cong T, as \HomR(S,T)≅k\Hom_R(S, T) \cong k in that case via the action of endomorphisms.[2]

Examples and constructions

Concrete examples

Simple modules over a field $ k $ are precisely the one-dimensional vector spaces over $ k $, which are isomorphic to $ k $ itself viewed as a left $ k $-module under the standard scalar multiplication action.[6] These modules have no proper nonzero submodules because any nonzero element generates the entire space via scalar multiples, and the only subspaces are $ {0} $ and $ k $. This example illustrates the basic case where the ring is commutative and a division ring, making all nonzero modules free and simple ones minimal-dimensional.[6] For principal ideal domains (PIDs), such as the integers $ \mathbb{Z} $, the simple modules take the form $ R / pR $, where $ R $ is the PID and $ p $ is a prime element generating a maximal ideal.[4] In the case of $ R = \mathbb{Z} $, these are the cyclic groups $ \mathbb{Z}/p\mathbb{Z} $ for prime $ p $, which admit no proper subgroups other than the trivial one due to the prime order.[4] Such modules highlight how simplicity arises from quotients by maximal ideals in domains with unique factorization, providing torsion examples distinct from the vector space case over fields.[4] Over the full matrix ring $ M_n(D) $, where $ D $ is a division ring and $ n \geq 1 $, the simple left modules are isomorphic to $ D^n $, equipped with the natural action of matrices acting on column vectors by left multiplication.[6] This module is simple because any nonzero subspace is invariant under all matrices only if it is the full space, as the action densely spans the endomorphisms by the density theorem, though here it follows directly from the ring's structure.[6] Up to isomorphism, this is the unique simple module, underscoring how noncommutative semisimple Artinian rings like matrix algebras have a single isomorphism class of simples.[6] In group algebras $ kG $, where $ G $ is a finite group and $ k $ is an algebraically closed field whose characteristic does not divide $ |G| $, the simple modules are exactly the irreducible representations of $ G $.[7] These arise as the indecomposable summands in the semisimple decomposition of any finite-dimensional module, by Maschke's theorem, and correspond to the simple left ideals in the regular representation.[7] For instance, the trivial representation is always simple, while others depend on the group's structure, such as the two-dimensional irreducible for the symmetric group $ S_3 $ over $ \mathbb{C} $. This connects module theory to classical representation theory, where simplicity equates to irreducibility.[7] For path algebras $ kQ $ of a quiver $ Q $ over a field $ k $, the simple modules are those corresponding to the vertices of $ Q $: for each vertex $ v $, there is a simple module $ S_v $ with underlying space $ k $ at $ v $ and zero at all other vertices, where paths (including loops) act by zero.[8] These modules have no proper submodules because any nonzero element at $ v $ spans the full one-dimensional space, and arrows from other vertices map to zero.[8] If $ Q $ has no oriented cycles, these exhaust all simples up to isomorphism, one per vertex, demonstrating how quiver representations encode module categories combinatorially.[8]

General constructions

Every simple left RR-module SS is isomorphic to R/IR/I for some maximal left ideal II of the ring RR. This construction arises because any nonzero element s∈Ss \in S generates SS as an RR-module, making SS cyclic, and the annihilator AnnR(s)={r∈R∣rs=0}\mathrm{Ann}_R(s) = \{ r \in R \mid r s = 0 \} forms a maximal left ideal, yielding the isomorphism S≅R/AnnR(s)S \cong R / \mathrm{Ann}_R(s). Thus, the simple left RR-modules correspond bijectively to the maximal left ideals of RR. In semisimple rings, simple modules admit a particularly explicit description as direct summands of the regular module RR_RR. Specifically, each simple left submodule of RR_RR is a minimal left ideal, and these are classified by the primitive idempotents ee of RR, where the summand eReR is simple if and only if ee is primitive. The Artin-Wedderburn theorem further refines this: a semisimple ring RR decomposes as R≅∏i=1kMni(Di)R \cong \prod_{i=1}^k M_{n_i}(D_i), where each DiD_i is a division ring and ni≥1n_i \geq 1; the simple left RR-modules then correspond to the unique (up to isomorphism) simple left DiD_i-module, extended to the ii-th matrix component as the space of column vectors. For semisimple Artinian rings, which coincide with Artinian semisimple rings by the Artin-Wedderburn theorem, the simple modules arise as the simple factors in this decomposition. Each simple left RR-module is isomorphic to the unique simple left module over one of the division ring components DiD_i, and the multiplicity nin_i determines the dimension of the corresponding minimal left ideals in RR. This provides a complete classification: the isomorphism classes of simple left RR-modules are in bijection with the division ring factors {D1,…,Dk}\{D_1, \dots, D_k\} in the Wedderburn decomposition of RR. In the category of left RR-modules, indecomposable injective modules often feature simple modules as their socle elements. An indecomposable injective module EE has a simple socle Soc(E)\mathrm{Soc}(E), which is the unique minimal submodule essential in EE, and this socle is a simple module that embeds into every nonzero submodule of EE. For rings where injective modules are well-understood, such as Artinian rings, each indecomposable injective is the injective hull of its simple socle, providing a construction of simple modules as the essential building blocks of injectives. Simple left RR-modules are classified via their annihilator ideals, which are precisely the primitive left ideals of RR. For a simple left RR-module S≅R/IS \cong R/I with maximal left ideal II, the annihilator AnnR(S)=I\mathrm{Ann}_R(S) = I is primitive, meaning II annihilates SS and no larger left ideal does so for a simple module. This correspondence establishes that the isomorphism classes of simple left RR-modules are determined by the primitive left ideals, each of which is maximal among the annihilators of simple submodules.

Structural theorems

Relation to composition series

A composition series of an RR-module MM is a finite chain of submodules 0=M0⊂M1⊂⋯⊂Mn=M0 = M_0 \subset M_1 \subset \cdots \subset M_n = M such that each successive quotient Mi/Mi−1M_i / M_{i-1} is a simple module.[9] Such a series exists if and only if MM satisfies both the ascending chain condition (Noetherian) and descending chain condition (Artinian) on submodules.[10] A simple module has a composition series of length 1, given by the chain 0⊂M0 \subset M, as it admits no proper nonzero submodules.[11] The Jordan-Hölder theorem states that for any module with a composition series, all such series have the same length, and the multisets of their simple factors are identical up to isomorphism and permutation.[11] This uniqueness highlights the role of simple modules as the atomic building blocks in the decomposition of modules into irreducible constituents. The composition length ℓ(M)\ell(M) of a module MM is defined as the number of simple factors in any of its composition series, providing an invariant measure of the module's "size" in terms of its simple components.[9] In the context of module theory, a chief series is a maximal chain of submodules with simple factors, which aligns directly with a composition series since all submodules are "normal" under the module action; this structure is analogous to chief series in group theory, where the factors are minimal normal subgroups.[9] Modules possessing a composition series—equivalently, those of finite length—are precisely the Artinian and Noetherian modules, wherein the simple modules appear as the irreducible constituents determining the module's structure.[10]

Jacobson density theorem

The Jacobson density theorem provides a fundamental characterization of the action of a ring on its faithful simple modules, linking primitive rings to dense subrings of endomorphism rings. Specifically, let RR be a ring and SS a faithful simple left RR-module. By Schur's lemma, the endomorphism ring D=EndR(S)D = \mathrm{End}_R(S) is a division ring. The ring RR acts densely on SS over DD, meaning that for any finite DD-linearly independent elements x1,…,xn∈Sx_1, \dots, x_n \in S and arbitrary elements y1,…,yn∈Sy_1, \dots, y_n \in S, there exists an element r∈Rr \in R such that rxi=yir x_i = y_i for all i=1,…,ni = 1, \dots, n. This density condition ensures that the module SS faithfully reflects the structure of RR, as the action of RR can approximate any desired linear transformation on finite-dimensional subspaces. A ring RR is primitive if and only if it admits a faithful simple left module SS, and in this case, RR embeds as a dense subring of the ring of DD-linear endomorphisms EndD(S)\mathrm{End}_D(S). This implies that simple modules over primitive rings embed the ring's action in a way that is "universal" on finite spans, allowing RR to act transitively and flexibly on SS. Consequently, every simple module over a primitive ring captures the essential structural properties of RR, facilitating the classification of such rings without finiteness assumptions. The theorem was introduced by Nathan Jacobson in his 1945 paper, generalizing earlier results on matrix rings over division rings by showing that primitive rings behave like dense operator rings even in infinite dimensions. A proof proceeds by induction on the number nn. For n=1n=1 (the base case), since SS is simple, the RR-submodule generated by the nonzero x1x_1 is all of SS, so there exists r∈Rr \in R such that rx1=y1r x_1 = y_1. For n>1n > 1, first find elements λi∈R\lambda_i \in R such that λixi≠0\lambda_i x_i \neq 0 and λixj=0\lambda_i x_j = 0 for j≠ij \neq i (using linear independence and simplicity), then apply the base case to map the λixi\lambda_i x_i to adjusted targets. This constructs the required rr.[12] As a corollary, if RR is a simple Artinian ring (satisfying the descending chain condition on ideals), then it is primitive with a finite-length faithful simple module, making the dense embedding surjective and yielding that RR is isomorphic to a matrix ring over a division ring—a generalization of Wedderburn's little theorem.
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