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Division ring
Division ring
Division ring
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In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring[1] in which every nonzero element a has a multiplicative inverse; that is, an element usually denoted a–1, such that a a–1 = a–1 a = 1. So, (right) division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b–1 ≠ b–1 a.

A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields.

Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".[5] In some languages, such as French, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field).

All division rings are simple. That is, they have no two-sided ideal besides the zero ideal and itself.

Algebraic structures
Group-like
Group theory
Ring-like
Ring theory
Lattice-like
Module-like
Algebra-like

Relation to fields and linear algebra

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All fields are division rings, and every non-field division ring is noncommutative. The best known example is the ring of quaternions. If one allows only rational instead of real coefficients in the constructions of the quaternions, one obtains another division ring. In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring;[6] every division ring arises in this fashion from some simple module.

Much of linear algebra may be formulated, and remains correct, for modules over a division ring D instead of vector spaces over a field. Doing so, one must specify whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. In particular, every module has a basis, and Gaussian elimination can be used. So, everything that can be defined with these tools works on division algebras. Matrices and their products are defined similarly.[citation needed] However, a matrix that is left invertible need not to be right invertible, and if it is, its right inverse can differ from its left inverse. (See Generalized inverse § One-sided inverse.)

Determinants are not defined over noncommutative division algebras. Most things that require this concept cannot be generalized to noncommutative division algebras, although generalizations such as quasideterminants allow some results[clarification needed] to be recovered.

Working in coordinates, elements of a finite-dimensional right module can be represented by column vectors, which can be multiplied on the right by scalars, and on the left by matrices (representing linear maps); for elements of a finite-dimensional left module, row vectors must be used, which can be multiplied on the left by scalars, and on the right by matrices. The dual of a right module is a left module, and vice versa. The transpose of a matrix must be viewed as a matrix over the opposite division ring Dop in order for the rule (AB)T = BTAT to remain valid.

Every module over a division ring is free; that is, it has a basis, and all bases of a module have the same number of elements. Linear maps between finite-dimensional modules over a division ring can be described by matrices; the fact that linear maps by definition commute with scalar multiplication is most conveniently represented in notation by writing them on the opposite side of vectors as scalars are. The Gaussian elimination algorithm remains applicable. The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is the dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same and define the rank of a matrix.

Division rings are the only rings over which every module is free: a ring R is a division ring if and only if every R-module is free.[7]

The center of a division ring is commutative and therefore a field.[8] Every division ring is therefore a division algebra over its center. Division rings can be roughly classified according to whether or not they are finite dimensional or infinite dimensional over their centers. The former are called centrally finite and the latter centrally infinite. Every field is one dimensional over its center. The ring of Hamiltonian quaternions forms a four-dimensional algebra over its center, which is isomorphic to the real numbers.

Examples

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  • As noted above, all fields are division rings.
  • The quaternions form a noncommutative division ring.
  • The subset of the quaternions a + bi + cj + dk, such that a, b, c, and d belong to a fixed subfield of the real numbers, is a noncommutative division ring. When this subfield is the field of rational numbers, this is the division ring of rational quaternions.
  • Let be an automorphism of the field . Let denote the ring of formal Laurent series with complex coefficients, wherein multiplication is defined as follows: instead of simply allowing coefficients to commute directly with the indeterminate , for , define for each index . If is a non-trivial automorphism of complex numbers (such as conjugation), then the resulting ring of Laurent series is a noncommutative division ring known as a skew Laurent series ring;[9] if σ = id then it features the standard multiplication of formal series. This concept can be generalized to the ring of Laurent series over any fixed field , given a nontrivial -automorphism .

Main theorems

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Wedderburn's little theorem: All finite division rings are commutative and therefore finite fields. (Ernst Witt gave a simple proof.)

Frobenius theorem: The only finite-dimensional associative division algebras over the reals are the reals themselves, the complex numbers, and the quaternions.

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Division rings used to be called "fields" in an older usage. In many languages, a word meaning "body" is used for division rings, in some languages designating either commutative or noncommutative division rings, while in others specifically designating commutative division rings (what we now call fields in English). A more complete comparison is found in the article on fields.

The name "skew field" has an interesting semantic feature: a modifier (here "skew") widens the scope of the base term (here "field"). Thus a field is a particular type of skew field, and not all skew fields are fields.

While division rings and algebras as discussed here are assumed to have associative multiplication, nonassociative division algebras such as the octonions are also of interest.

A near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.

See also

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Notes

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References

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Further reading

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Division ring

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A division ring, also known as a skew field, is a ring with multiplicative identity in which division is possible for every nonzero element, meaning every non-zero element has a two-sided . Unlike a field, the multiplication in a division ring need not be commutative, though addition is always commutative and both operations satisfy the usual ring axioms including associativity and distributivity. All fields, such as the real numbers R\mathbb{R} and complex numbers C\mathbb{C}, are division rings, but the converse holds only when multiplication is commutative. Notable non-commutative examples include the , a four-dimensional algebra over the reals discovered by in , where elements are of the form a+bi+cj+dka + bi + cj + dk with i2=j2=k2=ijk=−1i^2 = j^2 = k^2 = ijk = -1. Other examples encompass over fields like , such as (−1,p)Q(-1, p)_{\mathbb{Q}} for primes p≡3(mod4)p \equiv 3 \pmod{4}, and rings of twisted over division rings. Division rings play a central role in non-commutative algebra, , and the of simple artinian rings via the Artin-Wedderburn theorem, which decomposes them into matrix rings over division rings. A landmark result is Wedderburn's little theorem (1905), which states that every finite division ring is commutative and thus a field, implying no nontrivial finite non-commutative division rings exist. Over the reals, the only finite-dimensional associative division algebras are R\mathbb{R}, C\mathbb{C}, and H\mathbb{H}, as proven by Frobenius and later extended by theorems of Hurwitz and Adams on normed division algebras. These structures underpin applications in physics, such as rotations in three dimensions via quaternions, and in through finite fields.

Definition and Properties

Definition

A division ring, also known as a skew field, is a ring RR equipped with an 00 and a multiplicative identity 11, where R≠{0}R \neq \{0\}, and every nonzero element a∈Ra \in R possesses a unique two-sided multiplicative inverse b∈Rb \in R such that ab=ba=1ab = ba = 1. The structure satisfies the standard ring axioms, including associativity of both and , commutativity of , the existence of additive inverses, and distributivity of over . Crucially, the requirement for two-sided inverses for all nonzero elements ensures the absence of zero divisors, distinguishing division rings from general rings, which may lack such inverses or contain zero divisors. Unlike fields, which are commutative division rings, the multiplication in a division ring need not commute, allowing for noncommutative structures while preserving the invertibility property. This noncommutativity is often emphasized by alternative notations such as "skew field" or "noncommutative field."

Basic Properties

Division rings exhibit several fundamental algebraic properties that stem directly from their defining axioms. Foremost among these is the absence of zero divisors. Suppose a≠0a \neq 0 and b≠0b \neq 0 in a division ring RR, and assume ab=0ab = 0. Multiplying both sides on the left by a−1a^{-1} yields b=a−1(ab)=a−1⋅0=0b = a^{-1}(ab) = a^{-1} \cdot 0 = 0, a contradiction. Similarly, if ba=0ba = 0, then a=0⋅a−1=b⋅0⋅a−1=0a = 0 \cdot a^{-1} = b \cdot 0 \cdot a^{-1} = 0, again a contradiction. Thus, division rings have neither left nor right zero divisors. This lack of zero divisors positions division rings as domains in the non-commutative sense, where the product of any two nonzero elements is nonzero. When the multiplication is commutative, a division ring reduces to a field, which is precisely an with every nonzero element invertible. Division rings are simple rings, meaning they possess no nontrivial two-sided ideals. To see this, consider any nonzero two-sided ideal II of RR. Let a∈Ia \in I with a≠0a \neq 0. Then a−1a=1∈Ia^{-1} a = 1 \in I, and for any r∈Rr \in R, r=r⋅1∈Ir = r \cdot 1 \in I. Thus, I=RI = R. The set of nonzero elements R∖{0}R \setminus \{0\} forms a under the ring's multiplication operation. This follows immediately from the existence of inverses for each nonzero element, along with the associative and distributive properties inherited from the ring structure, ensuring closure, identity (the multiplicative unit 1), and inverses. Division rings support unique left and right division by nonzero elements. For any a≠0a \neq 0 and b∈Rb \in R, the equation ax=ba x = b has a unique solution x=a−1bx = a^{-1} b, as left multiplication by aa is injective (due to no zero divisors) and surjective (by solving for any right-hand side). Similarly, ya=by a = b has a unique solution y=ba−1y = b a^{-1}. The characteristic of a division ring is either zero or a pp. If the characteristic n>0n > 0 is composite, say n=kmn = k m with 1<k,m<n1 < k, m < n, then k⋅1=0k \cdot 1 = 0 and m⋅1=0m \cdot 1 = 0, implying zero divisors unless one is zero, which contradicts the domain property. Thus, nn must be prime. In characteristic zero, the ring contains an isomorphic copy of Z\mathbb{Z}.

History

Early Concepts

The roots of division ring concepts emerged in early 19th-century number theory, particularly through Carl Friedrich Gauss's exploration of algebraic integers. In his 1801 Disquisitiones Arithmeticae, Gauss introduced the Gaussian integers Z={a+bi∣a,b∈Z}\mathbb{Z} = \{a + bi \mid a, b \in \mathbb{Z}\}, demonstrating their unique factorization into primes and establishing them as an integral domain without zero divisors, which provided an early model for ring-like structures supporting division. Building on this, Ernst Kummer in the 1840s developed the notion of ideal numbers to address the breakdown of unique factorization in rings of cyclotomic integers, such as those arising from roots of unity. Kummer's 1844 and 1847 papers on ideal complex numbers treated "ideals" as abstract entities to restore a form of unique factorization, influencing later abstract approaches to divisibility in rings without zero divisors. The terminology for these structures began to solidify in the 1870s and 1880s through the work of Leopold Kronecker and Richard Dedekind. They employed the German term Körper ("body") for systems of numbers closed under addition, subtraction, multiplication, and division by nonzero elements, referring to what are now termed fields (commutative division rings). Noncommutative structures, such as quaternions, were recognized but treated separately in other contexts. Dedekind's 1871 supplements to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie marked a key formalization, defining the ring of integers in an algebraic number field as a structure without zero divisors and introducing ideals as multiplicative subsets to rigorously handle factorization and divisibility. This laid essential groundwork for division ring theory by abstracting ring properties from specific number fields. In the late 19th century, the analysis of polynomial rings in invariant theory further propelled these ideas, as mathematicians like Paul Gordan and David Hilbert examined ideals and syzygies in polynomial rings over fields to solve problems of invariant generation under group actions. Hilbert's 1890 basis theorem for ideals in polynomial rings exemplified how such structures without zero divisors facilitated broader algebraic insights. Hamilton's 1843 invention of quaternions offered an early concrete noncommutative example, extending complex numbers to a four-dimensional division ring.

Key Developments

In 1843, William Rowan Hamilton discovered the quaternions, marking the first explicit construction of a noncommutative division ring and inspiring subsequent efforts to identify higher-dimensional analogues over the real numbers. This breakthrough extended the complex numbers beyond two dimensions while preserving division properties, though multiplication proved noncommutative, challenging prevailing assumptions about algebraic structures. Building on this foundation, Ferdinand Georg Frobenius advanced the classification of finite-dimensional associative division algebras over the reals in his 1877-1878 works, demonstrating that only the reals, complexes, and quaternions satisfy the conditions. His analysis of composition algebras and bilinear forms provided early insights into the constraints on such structures, limiting possibilities to these three cases up to isomorphism and influencing later studies in normed algebras. A pivotal result came in 1905 when Joseph Henry Maclagan Wedderburn proved that every finite division ring is commutative, now known as Wedderburn's little theorem, which resolved a longstanding conjecture by showing such rings coincide with finite fields. This theorem, detailed in his paper on hypercomplex numbers, established a fundamental dichotomy between finite and infinite cases, with profound implications for the structure of finite rings. In 1927, Emil Artin extended Wedderburn's ideas through the Wedderburn-Artin theorem, which decomposes simple Artinian rings into matrix rings over division rings, providing a complete structural description for such algebras. This result unified earlier classifications and highlighted the role of division rings as building blocks in noncommutative ring theory. The 1930s saw significant progress in connecting division rings to broader algebraic frameworks, particularly through the work of Richard Brauer, Helmut Hasse, and Emmy Noether on central simple algebras and . Their collaborative efforts, culminating in the 1931 Brauer-Hasse-Noether theorem, established that central division algebras over number fields are cyclic and satisfy a local-global principle for splitting, linking these structures to class field theory via Hasse invariants. Noether's emphasis on algebraic invariants and Brauer's Sylow methods were instrumental in these advancements, solidifying the arithmetic theory of division rings. By the mid-20th century, terminology for noncommutative division rings standardized in English mathematical literature to "division ring" or "skew field" to distinguish them clearly from commutative fields and avoid earlier ambiguities like "hypercomplex field." This shift, as systematized in influential texts, facilitated precise discourse in ring theory and algebra.

Examples

Commutative Examples

Commutative division rings coincide precisely with the class of fields, since the commutativity of multiplication ensures that the standard division algorithm applies without obstruction. In such structures, every nonzero element admits a multiplicative inverse, and the ring is an integral domain. The prime fields serve as foundational examples. The rational numbers Q\mathbb{Q} form the prime field of characteristic 0, obtained as the field of fractions of the integers Z\mathbb{Z}. For each prime pp, the finite field Fp\mathbb{F}_p (also denoted Z/pZ\mathbb{Z}/p\mathbb{Z}) is the prime field of characteristic pp, consisting of the integers modulo pp under addition and multiplication. Algebraic extensions provide further commutative examples. The complex numbers C\mathbb{C} arise as a quadratic extension of the real numbers R\mathbb{R} by adjoining a root of x2+1=0x^2 + 1 = 0, yielding C=R\mathbb{C} = \mathbb{R} where i2=−1i^2 = -1. Finite fields also admit algebraic extensions: for a prime pp and positive integer nn, the field Fpn\mathbb{F}_{p^n} is the splitting field of the polynomial xpn−xx^{p^n} - x over Fp\mathbb{F}_p, containing pnp^n elements. The real numbers R\mathbb{R} stand out as the unique (up to isomorphism) ordered archimedean field, complete with respect to the absolute value metric. Transcendental extensions include function fields, such as the rational function field Q(x)\mathbb{Q}(x) over Q\mathbb{Q}, which consists of quotients of polynomials in one indeterminate xx. Local fields like the pp-adic numbers Qp\mathbb{Q}_p offer completions of Q\mathbb{Q} at the prime ideal (p)(p), forming a complete metric field of characteristic 0 with respect to the pp-adic valuation. These examples illustrate the breadth of commutative division rings, all fitting within the general framework of fields.

Noncommutative Examples

The most prominent example of a noncommutative division ring is the algebra of Hamilton's quaternions, denoted H\mathbb{H}, which is a 4-dimensional algebra over the real numbers R\mathbb{R} with basis {1,i,j,k}\{1, i, j, k\}. The defining relations are i2=j2=k2=−1i^2 = j^2 = k^2 = -1, ij=kij = k, and ji=−kji = -k, ensuring noncommutativity since ij=−jiij = -ji. Every nonzero quaternion q∈Hq \in \mathbb{H} is invertible, with the inverse given by q−1=qˉ∣q∣2q^{-1} = \frac{\bar{q}}{|q|^2}, where qˉ\bar{q} is the conjugate and the norm is ∣q∣=qqˉ|q| = \sqrt{q \bar{q}}
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