On the Zero Divisors and Extension of the Pythagorean Ring

Résumé

Background Let P denote the ring of the Pythagorean triples and Z(P) be its subset of zero divisors. It is well known that P is a commutative ring with identity ⟨ 3 , 4 , 5 ⟩ and is isomorphic to the ring ℤ 2 Method An enhanced understanding of the structure of P is assured by investigating its elements. Although the unit group of P is the Klein 4 group, there are infinitely many zero divisors of P and this paper explores the structure of the zero divisors of P . Although it is straightforward to identify the zero divisors of ℤ 2 , the identification of the zero divisors of P is quite laborious. Results In this paper, we have given a complete description of all the zero divisors of P . This is a significant contribution to the classification of Pythagorean triples which can be explored through the interplay between ring theoretic properties of P and graph theoretic properties of the zero divisor graph of P . Further, an extension of P has been constructed. The automorphisms of the extension ring have been investigated and its unit group characterized. Conclusion The findings on the zero divisors in P provide insights for further research, on the interplay between the theoretical properties of the ring P and the theoretical properties of the graph of Z ( P ) . Zero divisor graphs are ubiquitous models of both natural and man-made structures, and they find applications in communication networks, computer algorithms, and computational geometry. We have constructed an extension ring of P and established that its group of units R ∗ is isomorphic to ℤ 2 2 × ℤ h . Finally, we have determined its group of automorphisms, which may find applications in computer graphics, modeling, and designs.

Citer ce document

Accès au document

Texte intégral en lecture en ligne, réservé aux abonnés SPHAERO et aux membres de l'institution. Se connecter

Auteur(s)

Statistiques

Consultations : 1

Téléchargements : 0