On the Right Nucleus of Petit Algebras

Let $D$ be division algebra over its center $C$, let $\sigma$ be an endormorphism of $D$, let $\delta$ be a left $\sigma$-derivation of $D$, and let $R=D[t;\sigma ,\delta ]$ be a skew polynomial ring. We study the structure of a class of nonassociative algebras, denoted by $S_f$, whose construction canonically generalises that of the associative quotient algebras $R\mathrm{/}Rf$ where $f\in R$ is right-invariant. We determine the structure of the right nucleus of $S_f$ when the polynomial $f$ is bounded and not right invariant and either $\delta =0$, or $\sigma ={\mathrm{i}\mathrm{d}}_D$. As a by-product, we obtain a new proof on the size of the right nuclei of the cyclic (Petit) semifields ${\mathbb{S}}_f$. We look at subalgebras of the right nucleus of $S_f$, generalising several of Petit’s results \cite{petit1966certains} and introduce the notion of semi-invariant elements of the coefficient ring $D$. The set of semi-invariant elements is shown to be equal to the nucleus of $S_f$ when $f$ is not right-invariant. Moreover, we compute the right nucleus of $S_f$ for certain $f$. In the final chapter of this thesis we introduce and study a special class of polynomials in $R$ called generalised A-polynomials. In a differential polynomial ring over a field of characteristic zero, A-polynomials were originally introduced by Amitsur \cite{amitsur1954differential}. We find examples of polynomials whose eigenring is a central simple algebra over the field $C\cap \mathrm{F}\mathrm{i}\mathrm{x}(\sigma )\cap \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}(\delta )$.