The \emph{canonical structures of the plane} are those that result, up toisomorphism, 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 $\theta$ isthe equivalence class of x, which satisfies $\theta^2 = (-\dfrac{c}{a}) +\theta (-\dfrac{b}{a})$. On the other hand, it is known that, up toisomorphism, there are only three canonical structures: the corresponding to$\theta^2 = -1$ (the complex numbers), $\theta^2 = 1$ (the perplex orhyperbolic numbers) and $\theta^2 = 0$ (the parabolic numbers). This articlecopes with the algebraic structure of the rings of integers$\mathds{Z}[\theta]$ in the perplex and parabolic cases by \emph{analogy} tothe complex cases: the ring of Gaussian integers. For those rings a\emph{division algorithm} is proved and it is obtained, as a consequence, thecharacterization of the prime and irreducible elements.
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