Hyperideal-based zero-divisor graph of the general hyperring $ \mathbb{Z}_{n} $

Mohammad Hamidi; Irina Cristea; Mohammad Hamidi; Irina Cristea

doi:10.3934/math.2024768

AIMS Mathematics

2024, Volume 9, Issue 6: 15891-15910. doi: 10.3934/math.2024768

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Hyperideal-based zero-divisor graph of the general hyperring $ \mathbb{Z}_{n} $

Mohammad Hamidi ¹,
Irina Cristea ^{2
,
,}

1.
Department of Mathematics, Payame Noor University, Tehran, Iran
2.
Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, 5000, Nova Gorica, Slovenia

Received: 05 March 2024 Revised: 27 April 2024 Accepted: 28 April 2024 Published: 06 May 2024
MSC : 13A70, 16Y20

The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring $ R $ having a hyperideal $ I $, the $ I $-based zero-divisor graph $ \Gamma^{(I)}(R) $ associated with $ R $ is the simple graph whose vertices are the elements of $ R\setminus I $ having their hyperproduct in $ I $, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with $ I $. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the $ I $-based zero-divisor graph associated to the general hyperring $ \mathbb{Z}_n $ of the integers modulo $ n $, when $ n = 2p^mq $, with $ p $ and $ q $ two different odd primes, and fixing the hyperideal $ I $.
- general hyperring,
- hyperideal,
- absorbing elements,
- hyperideal-based zero-divisor graph
Citation: Mohammad Hamidi, Irina Cristea. Hyperideal-based zero-divisor graph of the general hyperring $ \mathbb{Z}_{n} $[J]. AIMS Mathematics, 2024, 9(6): 15891-15910. doi: 10.3934/math.2024768

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Abstract

The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring $ R $ having a hyperideal $ I $, the $ I $-based zero-divisor graph $ \Gamma^{(I)}(R) $ associated with $ R $ is the simple graph whose vertices are the elements of $ R\setminus I $ having their hyperproduct in $ I $, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with $ I $. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the $ I $-based zero-divisor graph associated to the general hyperring $ \mathbb{Z}_n $ of the integers modulo $ n $, when $ n = 2p^mq $, with $ p $ and $ q $ two different odd primes, and fixing the hyperideal $ I $.

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