nLab skewfield

Contents

Contents

Definition

A skewfield (also spelled skew-field), or reciprocal ring is a unital ring where each non-zero element has a two-sided inverse (but zero does not), or equivalently, an associative unital \mathbb{Z}-reciprocal algebra where the multiplicative identity element 11 is not zero.

A division ring is a unital ring where each non-zero element has a left inverse and a right inverse, or equivalently, an associative unital \mathbb{Z}-division algebra where the multiplicative identity element 11 is not zero. However in all associative unital division algebras, left and right inverses are the same, for the same reason that associative unital quasigroups and associative unital invertible magmas are both groups. As a result, division rings are the same as skewfields.

A commutative skewfield is called a field, but sometimes in specialized works on skewfields one often says simply field for skewfield.

In constructive mathematics and internally, the same issues appear for skewfields as for fields, and may be dealt with in the same way.

Linear algebra is often understood in the generality of division rings, namely the usual notions of linear basis, dimension, linear map, matrix of a linear map with respect to two bases and so on, and even Gauss elimination procedure, hold without changes for left or right vector spaces over a division rings.

Examples

The most famous noncommutative example is the skewfield of quaternions.

The Frobenius theorem states that apart from the fields of real and complex numbers and quaternions, there are no associative finite-dimensional division algebras over the real numbers; and even if one includes nonassociative finite-dimensional division algebras one obtains only one more example (the octonions). See at normed division algebra for more on this.

References

Last revised on June 12, 2022 at 13:19:02. See the history of this page for a list of all contributions to it.