On the right nucleus of Petit algebras

Let D be division algebra over its center C, let σ be an endormorphism of D, let δ be a left σ-derivation of D, and let R=D[t; σ, δ] be a skew polynomial ring. We study the structure of a class of nonassociative algebras, denoted by Sf, whose construction canonically generalises that of the associative quotient algebras R/Rf where f ∈ R is right-invariant. We determine the structure of the right nucleus of Sf when the polynomial f is bounded and not right invariant and either δ = 0, or σ = idD. As a by-product, we obtain a new proof on the size of the right nuclei of the cyclic (Petit) semifields Sf. We look at subalgebras of the right nucleus of Sf, generalising several of Petit's results [Pet66] 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 Sf when f is not right-invariant. Moreover, we compute the right nucleus of Sf 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 [Ami54]. We find examples of polynomials whose eigenring is a central simple algebra over the field C ∩ Fix(σ) ∩ Const(δ)