Algebras whose right nucleus is a central simple algebra

TL;DR

This paper extends classical algebra constructions to nonassociative algebras, demonstrating the existence of algebras with prescribed nuclei and analyzing their properties over various fields.

Contribution

It introduces a method to construct nonassociative algebras with a given central division algebra as their right nucleus, generalizing Amitsur's work to a broader algebraic context.

Findings

Existence of infinite-dimensional nonassociative algebras with prescribed nuclei.

Proof that certain p-algebras are cyclic differential extensions.

Construction of finite-dimensional division algebras with specified nuclei.

Abstract

We generalize Amitsur's construction of central simple algebras over a field $F$ which are split by field extensions possessing a derivation with field of constants $F$ to nonassociative algebras: for every central division algebra $D$ over a field $F$ of characteristic zero there exists an infinite-dimensional unital nonassociative algebra whose right nucleus is $D$ and whose left and middle nucleus are a field extension $K$ of $F$ splitting $D$, where $F$ is algebraically closed in $K$. We then give a short direct proof that every $p$-algebra of degree $m$, which has a purely inseparable splitting field $K$ of degree $m$ and exponent one, is a differential extension of $K$ and cyclic. We obtain finite-dimensional division algebras over a field $F$ of characteristic $p>0$ whose right nucleus is a division $p$-algebra.