## Abstract

An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.

Original language | English |
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Pages (from-to) | 35-84 |

Number of pages | 50 |

Journal | Journal of Algebra |

Volume | 649 |

DOIs | |

State | Published - 1 Jul 2024 |

### Bibliographical note

Publisher Copyright:© 2024 Elsevier Inc.

## Keywords

- Brauer group
- Central simple algebras
- Deformation of matrix algebras
- Maximal subfield
- Nonassociative algebra