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⊆M, the graph G(R;M,N), induced on R by the pair (M,N), is a simple graph with vertex set R∗=R∖{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 Zn-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

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Communications in Combinatorics and Optimization

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

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