  This paper is concerned with the construction of a small, but non-trivial,
example of a polynomial identity algebra, which we call the \emph{Jackson
algebra}, that will be used in sequels to this paper to study non-commutative
arithmetic geometry. In this paper this algebra is studied from a
ring-theoretic and geometric viewpoint. Among other things it turns out that
this algebra is a "non-commutative family" of central simple algebras and thus
parametrises Brauer classes over extensions of the base.
