  Let $S$ be a unital associative ring and $S[t;\sigma,\delta]$ be a skew
polynomial ring, where $\sigma$ is an injective endomorphism of $S$ and
$\delta$ a left $\sigma$-derivation. For each $f\in S[t;\sigma,\delta]$ of
degree $m>1$ with a unit as leading coefficient, we construct a unital
nonassociative algebra whose behaviour reflects the properties of $f$. The
algebras obtained yield canonical examples of right division algebras when $f$
is irreducible. We investigate the structure of these algebras. The structure
of their right nucleus depends on the choice of $f$. In the classical
literature, this nucleus appears as the eigenspace of $f$, and is used to
investigate the irreducible factors of $f$. We give necessary and sufficient
criteria for skew polynomials of low degree to be irreducible.
