Zero divisor graph of \mathbb{Z}_{2^{r} q^{s}}

• Juan M. Otero Acosta Departamento de Informática, Universidad Clodosbaldo Russián Cumaná, República Bolivariana de Venezuela. https://orcid.org/0009-0009-8245-9803
• Daniel Brito Departamento de Matemática, Universidad de Oriente Cumaná, República Bolivariana de Venezuela.
• Tobías de Jesús Rosas Soto Departamento de Matemática, Facultad Experimental de Ciencias, Universidad del Zulia, Maracaibo, Estado Zulia, República Bolivariana de Venezuela. https://orcid.org/0000-0002-8085-5011

Keywords: Rings, zero divisor set, zero dividers graph

Abstract

This article continues the study of zero divisor graphs, presented in 1988 by Istan Beck \cite{Beck}. There, a zero divisor graph is defined as a graph whose vertices are the elements of the set of zero divisors of a ring, where two distinct vertices $x$ and $y$ are adjacent if and only if $ x \cdot y = 0$. In this work, we present a new way to calculate the zero divisor graph of the ring $\mathbb{Z}_{2^{r}q^{s}}$ for $q$ an odd prime, with $r$ and $s$ positive integers greater than $2$, and the example of the zero divisor graph of the ring $\mathbb{Z}_{36}$ is also given.

References

[1] Andersen, D. and Livinston, P. The zero divisor graph of a conmutative ring, J. Algebra, 217 (1999), 434-447.
[2] Beck, I. Coloring of Conmutative rings, J. Algebra, 116 (1988), 288-226.
[3] Chartrand, G. and Lesniak, L. Graphs and Digraphs, Wadsworth and Brooks. 3era ed, California (1986).
[4] Fanelli, C. Grafo Divisor de Zero de un Anillo Conmutativo, Tesis de Maestrı́a, Universidad de Maringa, Brazil, (2011).
[5] Otero, J.Un método matricial para el cálculo de las constantes de Davenport y Olson k-baricéntricas. Tesis de Maestrı́a. Universidad de Oriente. Venezuela. 2011.
[6] Rojo, A. Álgebra I., Buenos Aires, Argentina, 1983.
[7] Shuker, N.; Mohammad, H. and Ali, A. The Zero Divisor Graph of Z p n q , International Journal of Algebra, 6 (2012), 1049-1055.
[8] Villarroel, F. La constante de Olson k-baricéntrica y un teorema inverso de Erds-Ginzburg-Ziv. Tesis Doctoral. Universidad Central de Venezuela. 2008.