The Ring of Integers in the Canonical Structures of the Planes

TL;DR

This paper explores the algebraic structure of rings of integers in canonical plane structures, establishing division algorithms and characterizing primes in perplex and parabolic cases by analogy to Gaussian integers.

Contribution

It introduces division algorithms and prime characterizations for rings of integers in perplex and parabolic structures, extending known complex number results.

Findings

Division algorithms are established for rings of integers in perplex and parabolic cases.

Prime and irreducible elements are characterized within these rings.

The approach uses analogy to Gaussian integers in complex numbers.

Abstract

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.