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Introduction to the arithmetic of the dual integers

The arithmetic structure of the special rings are of major interests for researchers in various fields. But until now, researchers are mainly focused on the arithmetic of the Euclidean and Dedeking domains. In this paper we study the arithmetic of the set $\mathbb{D}=\mathbb{Z}+\epsilon \mathbb{Z}$ with $\epsilon^{2}=0$ of dual integers which is not a domain because of the existence of divisors of zero. We introduce the notions of pseudo-factorial and pseudo-Euclidean ring. For dual integers, in general, the factorization and the remainder on pseudo-division are not unique because of the existence of zero divisors. Nevertheless, we have successfully exhibit an algorithm that output always the same remainder for pseudo-division. Hence many known algorithms on integers $\mathbb{Z}$ and on Gaussian integers $\mathbb{Z}[i]$ can be rewritten on dual integers $\mathbb{D}$. Similar results are obtained for $\frac{\mathbb{D}}{p\mathbb{D}}[x]$ where $p$ is a prime integer. In this paper all algorithms in $\mathbb{D}$, $\mathbb{D}[x]$ and $\frac{\mathbb{D}}{p\mathbb{D}}[x]$ are implemented in C/C++ using Victor Shoup's algebra system NTL.

Auteur(s) : Ghouraïssiou Camara and Mamadou; Sow, Djiby
Pages : 13–39
Année de publication : 2011
Revue : Far East J. Math. Sci. (FJMS)
N° de volume : 57
Type : Article
Mise en ligne par : SOW Djiby