Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2024; 61(3): 763-777

Online first article May 17, 2024      Printed May 31, 2024

https://doi.org/10.4134/BKMS.b230348

Copyright © The Korean Mathematical Society.

Rings whose associated extended zero-divisor graphs are complemented

Driss Bennis , Brahim El Alaoui , Raja L'hamri

Mohammed V University in Rabat; Mohammed V University in Rabat; Mohammed V University in Rabat

Abstract

Let $R$ be a commutative ring with identity $1\neq 0$. In this paper, we continue the study started in \cite{DJF} to further investigate when the extended zero-divisor graph of $R$, denoted as $\overline{\Gamma}(R)$, is complemented. We also study when $\overline{\Gamma}(R)$ is uniquely complemented. We give a complete characterization of when $\overline{\Gamma}(R)$ of a finite ring $R$ is complemented. Various examples are given using the direct product of rings and idealizations of modules.

Keywords: Commutative ring, zero-divisor graph, extended zero-divisor graph, complemented, uniquely complemented, zero-dimensional ring

MSC numbers: 13A70, 05C25, 05E40, 13B30