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

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 ${\mathrm{\Gamma }}^{(I)}(R)$ 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 $n=2p^{m}q$, with $p$ and $q$ two different odd primes, and fixing the hyperideal $I$.