The Ring of Integers in the Canonical Structures of the Planes
| dc.creator | Cifuente, José C. | |
| dc.creator | Strapasson, João E. | |
| dc.creator | Corrêa, Ana C. | |
| dc.creator | Kitani, Patrícia M. | |
| dc.date | 2007-07-05 | |
| dc.date.accessioned | 2026-07-07T08:14:05Z | |
| dc.date.available | 2026-07-07T08:14:05Z | |
| dc.description | 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 $\mathds{R}[θ]$, where $θ$ is the equivalence class of x, which satisfies $θ^2 = (-\dfrac{c}{a}) + θ(-\dfrac{b}{a})$. On the other hand, it is known that, up to isomorphism, there are only three canonical structures: the corresponding to $θ^2 = -1$ (the complex numbers), $θ^2 = 1$ (the perplex or hyperbolic numbers) and $θ^2 = 0$ (the parabolic numbers). This article copes with the algebraic structure of the rings of integers $\mathds{Z}[θ]$ 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. | |
| dc.description | 9 pages | |
| dc.identifier | https://arxiv.org/abs/0707.0700 | |
| dc.identifier | http://arxiv.org/abs/0707.0700 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/132995 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Commutative Algebra | |
| dc.subject | 13F99 | |
| dc.title | The Ring of Integers in the Canonical Structures of the Planes | |
| dc.type | text |