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Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. More formally, a ring with 0 ≠ 1 is a division ring if every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1). Division rings differ from fields only in that their multiplication is not required to be commutative. The condition 0 ≠ 1 is only there to exclude the trivial ring with a single element 0 = 1. Stated differently, a ring is a division ring if and only if the group of units is the set of all non-zero elements.

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 the endomorphism ring of S is a division ring; 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. 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 quaternion ring forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.

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 division algebras over the reals are the real numbers, the complex numbers and the quaternions.

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).

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.

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