The ring of integers in the canonical structures of the plane

Abstract

The 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 $\theta^2 = \left( -\dfrac{c}{a} \right) + \theta \left(-\dfrac{b}{a}\right)$. On the other hand, it is known that, up to isomorphism, there are only three canonical structures: the one 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 deals with the algebraic structure of the rings of integers $\mathds{Z}[\theta]$ in the perplex and parabolic cases by analogy to the complex cases: the ring of Gaussian integers. For those rings, a division algorithm is proved, and as a consequence, the characterization of the prime and irreducible elements is obtained.