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In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.

The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. It is then called a simple algebra over this field.

Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple.

Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring over a field does not have any nontrivial two-sided ideals (since any ideal of is of the form with an ideal of ), but it has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns).

An immediate example of a simple ring is a division ring, where every nonzero element has a multiplicative inverse, for instance, the quaternions. Also, for any , the algebra of matrices with entries in a division ring is simple.

Joseph Wedderburn proved that if a ring is a finite-dimensional simple algebra over a field , it is isomorphic to a matrix algebra over some division algebra over . In particular, the only simple rings that are finite-dimensional algebras over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions.

Wedderburn proved these results in 1907 in his doctoral thesis, On hypercomplex numbers, which appeared in the Proceedings of the London Mathematical Society. His thesis classified finite-dimensional simple and also semisimple algebras over fields. Simple algebras are building blocks of semisimple algebras: any finite-dimensional semisimple algebra is a Cartesian product, in the sense of algebras, of finite-dimensional simple algebras.

One must be careful of the terminology: not every simple ring is a semisimple ring, and not every simple algebra is a semisimple algebra. However, every finite-dimensional simple algebra is a semisimple algebra, and every simple ring that is left- or right-artinian is a semisimple ring.

Wedderburn's result was later generalized to semisimple rings in the Wedderburn–Artin theorem: this says that every semisimple ring is a finite product of matrix rings over division rings. As a consequence of this generalization, every simple ring that is left- or right-artinian is a matrix ring over a division ring.

Examples

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Let be the field of real numbers, be the field of complex numbers, and the quaternions.

  • A central simple algebra (sometimes called a Brauer algebra) is a simple finite-dimensional algebra over a field whose center is .
  • Every finite-dimensional simple algebra over is isomorphic to an algebra of matrices with entries in , , or . Every central simple algebra over is isomorphic to an algebra of matrices with entries or . These results follow from the Frobenius theorem.
  • Every finite-dimensional simple algebra over is a central simple algebra, and is isomorphic to a matrix ring over .
  • Every finite-dimensional central simple algebra over a finite field is isomorphic to a matrix ring over that field.
  • Over a field of characteristic zero, the Weyl algebra is simple but not semisimple, and in particular not a matrix algebra over a division algebra over its center; the Weyl algebra is infinite-dimensional, so Wedderburn's theorem does not apply to it.

See also

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References

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Simple ring

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In ring theory, a simple ring is a nonzero ring that has no two-sided ideals other than the zero ideal and the ring itself. This property makes simple rings fundamental building blocks in the study of more complex ring structures, as they lack nontrivial invariant subspaces under the ring's multiplication action. Commutative simple rings are precisely the fields, since any nonzero ideal in a commutative ring is the entire ring if it contains a unit element. More generally, division rings—noncommutative rings in which every nonzero element has a multiplicative inverse—are simple. Finite matrix rings over division rings, denoted Mn(D)M_n(D) for a division ring DD and integer n≥1n \geq 1, also form simple rings, providing key noncommutative examples. The Weyl algebra over a field, which models differential operators, is another notable simple ring that is not semisimple. A cornerstone result classifying simple rings with additional finiteness conditions is the Wedderburn-Artin theorem, which states that a simple left or right is isomorphic to a Mn(D)M_n(D) over a DD, with nn and DD unique up to . In this context, the ring's modules decompose into direct sums of copies of the irreducible module DnD^n. Simple rings thus play a central role in semisimple theory, where broader structures decompose into direct products of such over .

Definition and basics

Formal definition

A simple ring is a nonzero ring RR that has no two-sided ideals other than the zero ideal {0}\{0\} and the ring itself RR. A two-sided ideal II of RR is an additive subgroup of RR such that for all r∈Rr \in R and i∈Ii \in I, both ri∈Iri \in I and ir∈Iir \in I. The condition that RR is nonzero ensures the exclusion of the trivial ring, where the only ideal coincides with zero and the ring itself, rendering the notion of simplicity vacuous. A commutative simple ring with a multiplicative identity is a field.

Relation to ideals

In , ideals play a fundamental role analogous to normal subgroups in group theory, serving as the kernels of ring and enabling the construction of . Specifically, a two-sided ideal II of a ring RR is the kernel of a surjective ϕ:R→S\phi: R \to S, where the R/IR/I is isomorphic to SS, mirroring how normal subgroups facilitate quotient groups. The absence of nontrivial two-sided ideals in a simple ring implies that it admits no proper nontrivial quotient rings, rendering the ring indecomposable under homomorphic images beyond the zero ring and itself. This structural rigidity underscores the simplicity condition as a marker of minimal ideal lattice, where the only ideals are the zero ideal and the entire ring. The concept of ideals originated with Richard Dedekind in 1871, who introduced them in the context of the ring of integers of algebraic number fields to resolve failures of unique factorization. David Hilbert extended the study of ideals to polynomial rings in 1893, proving key finiteness properties that advanced commutative algebra. Simplicity represents the extreme case of this ideal theory, embodying a ring with the sparsest possible two-sided ideal structure. While rings are defined by the absence of nontrivial two-sided ideals, they may possess nontrivial one-sided ideals, distinguishing the bilateral condition from unilateral absorption properties. As a brief note, this aligns with the earlier observation that commutative rings with a multiplicative coincide with fields.

Properties

Basic properties

rings are typically studied in the context of unital rings, where they possess a multiplicative identity element, denoted 11, distinct from the zero element 00. While non-unital rings exist in non-standard settings, they are pathological and not conventional. Simple rings contain no nonzero nilpotent ideals. Any nilpotent ideal II of RR satisfies Ik={0}I^k = \{0\} for some positive integer k>1k > 1, and if I≠{0}I \neq \{0\}, then II is a proper two-sided ideal, contradicting the simplicity of RR. Thus, the only nilpotent ideal is the zero ideal. Every simple ring is a prime ring. Specifically, for any two-sided ideals A,B⊆RA, B \subseteq R, if AB={0}AB = \{0\}, then either A={0}A = \{0\} or B={0}B = \{0\}, as the product ABAB forms a two-sided ideal contained in the zero ideal. This property underscores the absence of zero-divisor ideals in simple rings. The Jacobson radical J(R)J(R) of a simple ring RR, defined as the intersection of all maximal left ideals of RR, is the zero ideal. Since J(R)J(R) is itself a two-sided ideal and RR admits no proper nonzero two-sided ideals, J(R)J(R) must be trivial. This implies that RR is semisimple in the Jacobson sense. In a simple ring RR with unity, for any nonzero element a∈Ra \in R, the two-sided ideal generated by aa, denoted RaRRaR, equals RR. Indeed, RaRRaR is a two-sided ideal containing a⋅1⋅a=a2≠{0}a \cdot 1 \cdot a = a^2 \neq \{0\} (as nilpotency would imply a nilpotent ideal), and thus must coincide with the entire ring RR. This property highlights the "generative" nature of nonzero elements in simple rings.

Center and simple modules

The center of a simple ring RR, denoted Z(R)Z(R), consists of all elements z∈Rz \in R such that zr=rzzr = rz for every r∈Rr \in R. This set forms a commutative subring of RR. Since RR is simple, any nonzero ideal of Z(R)Z(R) would generate a nonzero two-sided ideal of RR contained in the , which contradicts simplicity unless the ideal is all of Z(R)Z(R). Thus, Z(R)Z(R) has no proper nonzero ideals and is a field (assuming RR is unital, as is standard for simple rings). The ring RR becomes an algebra over the field Z(R)Z(R) via the natural embedding of Z(R)Z(R) into RR, where elements of Z(R)Z(R) act by left (or right) multiplication. As a left Z(R)Z(R)-module, RR is simple because any nonzero Z(R)Z(R)-submodule of RR would be a nonzero two-sided ideal of RR. Thus, RR is a simple algebra over its center Z(R)Z(R). The centralizer of RR in its endomorphism ring coincides with Z(R)Z(R), reinforcing that the center fully captures the commutative structure within simple rings. Every simple left RR-module MM is faithful, meaning its annihilator AnnR(M)={r∈R∣rm=0 ∀m∈M}\mathrm{Ann}_R(M) = \{ r \in R \mid rm = 0 \ \forall m \in M \} is zero. Indeed, AnnR(M)\mathrm{Ann}_R(M) is a two-sided ideal of RR; since MM is nonzero and simple, this ideal must be proper unless M=0M = 0, so simplicity of RR forces AnnR(M)=0\mathrm{Ann}_R(M) = 0. The same holds for simple right RR-modules.

Characterizations

Artinian simple rings

A left Artinian simple ring is a simple ring that satisfies the descending chain condition (DCC) on left ideals, meaning that every descending chain of left ideals stabilizes after finitely many steps. Equivalently, every nonempty collection of left ideals has a minimal element. This condition applies to the ring considered as a left module over itself, where submodules correspond to left ideals. In contrast to general simple rings, which may lack this finiteness property, Artinian simple rings exhibit controlled structure on one-sided ideals. Since a simple ring has only two two-sided ideals (the zero ideal and the ring itself), it trivially satisfies the DCC on two-sided ideals, with no possibility of infinite descending chains. However, the left Artinian condition imposes finiteness on the lattice of left ideals. By the Akizuki––Levitzki , every left Artinian ring has finite as a left module over itself, meaning it admits a finite with simple factors. Thus, Artinian simple rings have finite , ensuring that their left ideal structure terminates in a finite number of steps. This finite property highlights their bounded complexity compared to non-Artinian simple rings. Furthermore, every Artinian simple ring is semisimple as a left module over itself, decomposing as a finite of simple left modules. Semisimplicity here implies that the ring is a of its simple submodules, with the Jacobson radical vanishing. This semisimple nature follows from the primitive property of simple rings combined with the Artinian condition, which forces the absence of ideals and ensures projective simple modules. As a brief connection to module , the simple left modules over such rings are precisely the minimal left ideals.

Wedderburn–Artin theorem

The classifies the structure of Artinian simple rings, stating that every left Artinian simple ring RR is isomorphic to the matrix ring Mn(D)M_n(D) over a DD, where n≥1n \geq 1 is an . Equivalently, R≅End⁡D(V)R \cong \operatorname{End}_D(V) for some finite-dimensional left VV over the division ring DD. This theorem provides the foundational structure theory for such rings, showing they are precisely the endomorphism rings of finite-dimensional modules over division rings. The result originated with Joseph Henry Maclagan Wedderburn's 1908 work on finite-dimensional simple algebras over fields, where he proved that such algebras are matrix rings over division algebras. Wedderburn's proof relied on the decomposition of semisimple algebras into simple components and the analysis of their minimal ideals. In 1927, Emil Artin extended this to the general Artinian setting, incorporating the descending chain condition on left ideals without assuming a base field, thus establishing the theorem in its modern form. Artin's generalization highlighted the role of the Artinian condition in ensuring finite decompositions and division endomorphisms. A sketch of the proof begins by viewing RR as a faithful semisimple left module over itself. Since RR is simple and Artinian, every nonzero submodule is faithful, and RR decomposes as a finite of isomorphic simple left RR-modules, say R≅V⊕nR \cong V^{\oplus n} for a simple module VV. The ring End⁡R(V)op\operatorname{End}_R(V)^{\mathrm{op}} must then be a DD, as any nonzero is invertible due to the simplicity of VV and the Artinian condition preventing zero divisors in the endomorphisms. By the density theorem or in this context, RR acts as the full ring End⁡D(V)\operatorname{End}_D(V), yielding the isomorphism R≅Mn(D)R \cong M_n(D). This argument leverages the from the Artinian simple hypothesis, ensuring the decomposition is finite and the ring divides properly. A key applies to finite-dimensional over a field kk: every simple Artinian kk- is isomorphic to Mn(D)M_n(D), where DD is a central division kk- (i.e., of DD is exactly kk). This follows directly from the general theorem, with the centrality arising because the opposite endomorphism ring inherits the commutant as kk. The thus unifies the study of simple rings under the Artinian assumption, with profound implications for and noncommutative .

Examples

Division rings

A division ring, also known as a , is a nontrivial ring with multiplicative identity in which every nonzero element admits a two-sided , allowing by any nonzero element. When is commutative in a division ring, it is called a field. Every is a simple ring. To see this, suppose II is a nonzero two-sided ideal of a DD. Let a∈Ia \in I with a≠0a \neq 0; then a−1a=1∈Ia^{-1} a = 1 \in I, so I=DI = D. Thus, the only two-sided ideals are {0}\{0\} and DD itself. Examples of division rings include the fields of real numbers R\mathbb{R} and complex numbers C\mathbb{C}, which are commutative. A noncommutative example is the quaternion algebra H\mathbb{H}, discovered by on October 16, 1843, while walking along Dublin's , where he formulated the relations i2=j2=k2=ijk=−1i^2 = j^2 = k^2 = ijk = -1. The center of H\mathbb{H} consists precisely of the real scalars R\mathbb{R}. Wedderburn's little theorem states that every finite is commutative and hence a field. This result, proved by Wedderburn in , implies that examples of finite division rings are precisely the finite fields, such as Z/pZ\mathbb{Z}/p\mathbb{Z} for prime pp. Division rings serve as the basic building blocks in the , which decomposes finite-dimensional simple algebras over fields into matrix rings over such structures.

Matrix rings over division rings

The matrix ring Mn(D)M_n(D), where n≥1n \geq 1 is a positive integer and DD is a , consists of all n×nn \times n matrices with entries in DD, equipped with the standard operations of and . This construction generalizes the familiar matrix algebras over fields, extending to non-commutative settings where DD may not commute. As a left vector space over DD, Mn(D)M_n(D) has dimension n2n^2, with the standard matrix units EijE_{ij} (matrices with a 1 in the (i,j)(i,j)-entry and zeros elsewhere) forming a basis. A concrete example is Mn(R)M_n(\mathbb{R}), the ring of n×nn \times n real matrices, which is simple and non-commutative for n>1n > 1. If DD is a field (hence commutative), the of Mn(D)M_n(D) consists precisely of the scalar matrices λIn\lambda I_n for λ∈D\lambda \in D. To establish simplicity, consider any nonzero two-sided ideal II of Mn(D)M_n(D). The column spaces Lj={A∈Mn(D)∣AL_j = \{ A \in M_n(D) \mid A has zeros except possibly in the jj-th column}\} for j=1,…,nj = 1, \dots, n form minimal left ideals, each simple as a left Mn(D)M_n(D)-module. Specifically, for a nonzero A∈LjA \in L_j, left multiplication by suitable matrices generates all of LjL_j, including the matrix units, and thus II contains a full set of matrix units, generating the entire ring. Hence, I=Mn(D)I = M_n(D), proving that the only two-sided ideals are {0}\{0\} and Mn(D)M_n(D) itself. Matrix rings were first systematically studied by in his 1858 memoir, where he introduced the of square matrices and their . Their classification as simple rings follows from the , as established by J.H.M. Wedderburn in 1908.

Extensions

Non-Artinian simple rings

Simple rings are not necessarily Artinian. A canonical example is the ring of differential operators on the affine line over a field of characteristic zero, known as the first Weyl algebra A1(k)=k⟨x,∂⟩/(∂x−x∂−1)A_1(k) = k\langle x, \partial \rangle / (\partial x - x\partial - 1), where xx corresponds to by the coordinate and ∂\partial to differentiation. This ring is simple, as it has no nontrivial two-sided ideals, but it is not Artinian, since it contains infinite descending chains of left ideals. The Weyl A1(k)A_1(k) is both left and right Noetherian with global dimension 1, yet its modules lack finite due to the absence of the descending chain condition on submodules. In contrast to Artinian simple rings, which decompose as matrix rings over division rings, the Weyl exhibits more intricate module theory without finite-length decompositions. More generally, rings of differential operators on smooth affine varieties over fields of characteristic zero are simple non-Artinian rings, extending the Weyl construction to higher dimensions. Other examples include universal enveloping algebras of certain s that yield simple quotients, such as the Weyl algebra arising from the Heisenberg , and generalized Weyl algebras associated with simple s like sl(2,k)\mathfrak{sl}(2,k). These rings have infinite global dimension in higher cases and lack finite for modules. The fails to characterize non-Artinian simple rings, creating a gap in the classical structure theory. Their analysis often relies on filtrations, such as the order filtration on differential operator rings, where the associated graded algebra is commutative , facilitating the study of associated graded modules and filtrations on ideals.

Central simple algebras

A central simple algebra over a field kk is a finite-dimensional kk-algebra that is simple (having no nontrivial two-sided ideals) and central (having center equal to kk). Such algebras have dimension over kk that is a perfect square. By the , every AA over kk is isomorphic to a matrix algebra Mn(D)M_n(D), where n≥1n \geq 1 is an and DD is a over kk (finite-dimensional, simple, with center kk, and no zero divisors). The nn and the DD (up to ) are uniquely determined by AA. The Brauer group Br(k)\mathrm{Br}(k) of the field kk is the formed by the equivalence classes of central simple kk-s under Brauer equivalence, where two algebras AA and BB are equivalent if A⊗kB≅Mm(C)A \otimes_k B \cong M_m(C) for some m≥1m \geq 1 and some central simple kk- CC. The group operation is induced by the over kk, and Br(k)\mathrm{Br}(k) is a torsion group. For a central simple kk- AA, its class [A][A] in Br(k)\mathrm{Br}(k) is equal to [D][D], where DD is the central division kk- such that A≅Mn(D)A \cong M_n(D). Quaternion algebras provide concrete examples of central simple algebras. A quaternion algebra over Q\mathbb{Q} is a central simple Q\mathbb{Q}-algebra of dimension 4, typically denoted (a,b∣Q)(a,b \mid \mathbb{Q}) for a,b∈Q×a, b \in \mathbb{Q}^\times, with basis {1,i,j,k}\{1, i, j, k\} satisfying i2=ai^2 = a, j2=bj^2 = b, and ij=−ji=kij = -ji = k. For instance, Hamilton's quaternions (−1,−1∣Q)(-1, -1 \mid \mathbb{Q}) form a division algebra (nonsplit over R\mathbb{R}) with class of order 2 in Br(Q)\mathrm{Br}(\mathbb{Q}). Cyclic algebras offer further examples: given a cyclic Galois extension E/kE/k of degree nn with Galois group generated by σ\sigma, and b∈k×b \in k^\times, the cyclic algebra (E,σ,b)(E, \sigma, b) is the kk-vector space ⨁i=0n−1Eμi\bigoplus_{i=0}^{n-1} E \mu^i with multiplication μx=σ(x)μ\mu x = \sigma(x) \mu for x∈Ex \in E and μn=b\mu^n = b; this is a central simple kk-algebra of degree nn. The theory of central simple algebras was developed in the late 1920s and 1930s, with Richard Brauer playing a pivotal role through his work on the arithmetic properties of group representations and hypercomplex systems, including theorems on splitting fields and the structure of . Brauer's contributions culminated in the 1931–1932 collaboration with and , proving that every central division algebra over a number field is cyclic, which established the local-global principle for such algebras. This framework found applications in , notably in Hasse's 1933 proof of Artin's using the structure of Brauer groups to characterize abelian extensions. For a central simple kk-algebra A≅Mn(D)A \cong M_n(D) with central division kk-algebra DD, the period of AA is the order of its class [A][A] in Br(k)\mathrm{Br}(k), the smallest positive integer mm such that A⊗m≅Mr(k)A^{\otimes m} \cong M_r(k) for some rr, while the index of AA is the degree of DD, equal to dim⁡kD\sqrt{\dim_k D}
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