  The \emph{canonical structures of the plane} are those that result, up to
isomorphism, from the rings that have the form $\mathds{R}[x]/(ax^2+bx+c)$ with
$a\neq 0$.That ring is isomorphic to $\mathds{R}[\theta]$, where $\theta$ is
the equivalence class of x, which satisfies $\theta^2 = (-\dfrac{c}{a}) +
\theta (-\dfrac{b}{a})$. On the other hand, it is known that, up to
isomorphism, there are only three canonical structures: the corresponding to
$\theta^2 = -1$ (the complex numbers), $\theta^2 = 1$ (the perplex or
hyperbolic numbers) and $\theta^2 = 0$ (the parabolic numbers). This article
copes with the algebraic structure of the rings of integers
$\mathds{Z}[\theta]$ in the perplex and parabolic cases by \emph{analogy} to
the complex cases: the ring of Gaussian integers. For those rings a
\emph{division algorithm} is proved and it is obtained, as a consequence, the
characterization of the prime and irreducible elements.
