Generalized Beck’s Zero-Divisor Graph: A Graph Associated with a ring induced by a module-submodule pair

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Document Type : Original paper

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Department of Basic Sciences, Princess Sumaya University for Technology, Amman, Jordan

Abstract

Given a commutative ring $R$, a left $R$-module $M$, and an $R$-submodule $N \subseteq  M$, the graph $G(R;M,N)$, induced on $R$ by the pair $(M,N)$, is a simple graph with vertex set $R^* = R \backslash \{0\}$. Distinct vertices r and s are adjacent if $rsN = 0$. This graph generalizes Beck's zero-divisor graph $G(R)$. We analyze connectivity, completeness, bipartiteness, cycles, diameter, girth, independence/clique/chromatic numbers, and domination numbers, often under specific algebraic constraints on $R$ or $N$. Applications to $\mathbb Z_n$-modules illustrate these results. By linking $G(R;M,N)$ to $G(R)$, we derive graph invariants for $G(R)$ efficiently and vice versa, deepening insights into algebraic structures and their graph-theoretic analogs.

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References

[1] M. Ahmed and F. Moh’d, A new intersection-graph type for modules, Comm. Algebra 52 (2024), no. 5, 2065–2078.
https://doi.org/10.1080/00927872.2023.2280716
[2] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447.
https://doi.org/10.1006/jabr.1998.7840
[3] F.W. Anderson and K.R. Fuller, Rings and Categories of Modules, Springer Science & Business Media, 1992.
[4] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226.  https://doi.org/10.1016/0021-8693(88)90202-5
[5] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011), no. 4, 727–739.  https://doi.org/10.1142/S0219498811004896
[6] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer Publishing Company, Incorporated, 2008.
[7] H.R. Maimani, M. Salimi, A. Sattari, and S. Yassemi, Comaximal graph of commutative rings, J. Algebra 319 (2008), no. 4, 1801–1808.  https://doi.org/10.1016/j.jalgebra.2007.02.003
[8] F. Moh’d and M. Ahmed, A simple-intersection graph of a ring approach to solving coloring optimization problems, Commun. Comb. Optim. 10 (2025), no. 2, 423–442.  https://doi.org/10.22049/cco.2023.28858.1752
[9] F. Moh’d and M. Ahmed, Simple-intersection graphs of rings, AIMS Math. 8 (2023), no. 1, 1040–1054. https://doi.org/10.3934/math.2023051
[10] D. Sinha and B. Kaur, On Beck’s zero-divisor graph, Notes Number Theory Discrete Math. 25 (2019), no. 4, 150—-157.
[11] Y. Tian and L. Li, Comments on the clique number of zero-divisor graphs of $\mathbb{Z}_n$, J. Math. 2022 (2022), no. 1, Article ID: 6591317. https://doi.org/10.1155/2022/6591317
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 04 May 2025

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