Division ring

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In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i.e. an element x with ax = xa = 1). If rings are viewed as categorical constructions, then this is equivalent to requiring that all nonzero morphisms are isomorphisms. Division rings are very similar to fields except 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 iff 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 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 Gauss-Jordan elimination algorithm remains applicable.

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 states that all finite division rings are commutative and therefore finite fields.

Division rings used to be called fields in an older usage, which remained in other languages. A more complete comparison is found in the article Field (mathematics).

Skew fields have an interesting semantic feature: a prefix, here "skew", widens the scope of the suffix (here "field"). Thus a field is a particular type of skew field. This phenomenon appears to be rare in English, the only other example being Godemont's claim that tea is a particular kind of "leaf tea".de:Schiefkörper nl:Delingsring (Ned) / lichaam (Be) zh:除环