Revista de la
Unión Matemática Argentina
Genus and book thickness of reduced cozero-divisor graphs of commutative rings
Edward Jesili, Krishnan Selvakumar, and Thirugnanam Tamizh Chelvam

Volume 67, no. 2 (2024), pp. 455–473    

Published online (final version): September 23, 2024

https://doi.org/10.33044/revuma.3906

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Abstract

For a commutative ring $R$ with identity, let $\langle a\rangle$ be the principal ideal generated by $a\in R$. Let $\Omega(R)^*$ be the set of all nonzero proper principal ideals of $R$. The reduced cozero-divisor graph $\Gamma_r(R)$ of $R$ is the simple undirected graph whose vertex set is $\Omega(R)^*$ and such that two distinct vertices $\langle a\rangle$ and $\langle b\rangle$ in $\Omega(R)^\ast$ are adjacent if and only if $\langle a \rangle\nsubseteq\langle b\rangle$ and $\langle b\rangle\nsubseteq\langle a\rangle$. In this article, we study certain properties of embeddings of the reduced cozero-divisor graph of commutative rings. More specifically, we characterize all Artinian nonlocal rings whose reduced cozero-divisor graph has genus two. Also we find the book thickness of the reduced cozero-divisor graphs which have genus at most one.

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