The Ring of Integers in the Canonical Structures of the Planes

The \emph{canonical structures of the plane} are those that result, up to isomorphism, from the rings that have the form $\text{mathds}R[x]\mathrm{/}(a{x}^2+bx+c)$ with $a\mathrm{\ne }0$.That ring is isomorphic to $\text{mathds}R[\theta ]$, 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 $\text{mathds}Z[\theta ]$ 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.