Division ring

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In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring[1] in which every nonzero element a has a multiplicative inverse, i.e., an element x with a·x = x·a = 1. Stated differently, a ring is a division ring if and only if the group of units equals the set of all nonzero elements.

Division rings differ from fields only in that their multiplication is not required to be commutative. However, by Wedderburn's little theorem 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”.

Relation to fields and linear algebra[edit]

All fields are division rings; more interesting examples are the non-commutative division rings. The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain 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;[2] every division ring arises in this fashion from some simple module.

Much of linear algebra may be formulated, and remains correct, for (left) modules over division rings instead of vector spaces over fields. Every module over a division ring has a basis; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable. Differences between linear algebra over fields and skew fields occur whenever the order of the factors in a product matters. For example, the proof that the column rank of a matrix over a field equals its row rank yields for matrices over division rings only that the left column rank equals its right row rank: it does not make sense to speak about the rank of a matrix over a division ring.

The center of a division ring is commutative and therefore a field.[3] 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, of course, one-dimensional over its center. The ring of Hamiltonian quaternions forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.

Examples[edit]

  • As noted above, all fields are division rings.
  • The real and rational quaternions are strictly noncommutative division rings.
  • Let \sigma:\mathbb{C}\rightarrow\mathbb{C} be an automorphism of the field \mathbb{C}. Let \mathbb{C}((z,\sigma)) 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 z, for \alpha\in\mathbb{C}, define z^i\alpha:=\sigma^i(\alpha)z^i for each index i\in\mathbb{Z}. If \sigma is a non-trivial automorphism of complex numbers (such as the conjugation), then the resulting ring of Laurent series is a strictly noncommutative division ring known as a skew Laurent series ring;[4] 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 F, given a nontrivial F-automorphism \sigma.

Ring theorems[edit]

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.

Related notions[edit]

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 non-commutative 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 Field (mathematics).

Skew fields have 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.

Notes[edit]

  1. ^ In this article, rings have a 1.
  2. ^ Lam (2001), p. 33.
  3. ^ Simple commutative rings are fields. See Lam (2001), p. 39 and exercise 3.4 on p.45.
  4. ^ Lam (2001), p. 10

See also[edit]

References[edit]

  • Lam, Tsit-Yuen (2001). A first course in noncommutative rings. Graduate texts in mathematics 131 (2 ed.). Springer. ISBN 0-387-95183-0. 

External links[edit]