division algebra

Noun

Meaning

(algebra) A linear algebra such that its non-zero vectors form a multiplicative group.
Synonym: associative division algebra
Hypernyms: associative algebra, linear algebra, division ring, vector space
Hyponyms: field, quaternions, complex numbers, reals

(algebra) An algebra over a field such that every non-zero element of it has a multiplicative inverse. (It is not required to have a unity element.)
Hypernyms: algebra over a field, ring, vector space
Hyponym: octonions

Synonyms

Examples

A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. Source: Internet

By the Artin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the reals. Source: Internet

For example: * Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions. Source: Internet

Hence, the only complex Banach algebra which is a division algebra is the complexes. Source: Internet

Let A be a complex unital Banach algebra in which every non-zero element x is invertible (a division algebra). Source: Internet

Quaternions form a division algebra —every non-zero element has an inverse—whereas Pauli matrices do not. Source: Internet

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