Explanipedia

A generalization of <i>n</i>-ary prime subhypermodule Open

M. J. Nikmehr, R. Nikandish, A. Yassine, Kostaq Hila, Šárka Hošková-Mayerová · 2024

Let ( M, f, g ) be an ( m, n )-hypermodule over an ( m, n )-hyperring ( R, h, k ). A proper subhypermodule N of M is called n -ary 2-absorbing subhypermodule if whenever g ( r 1 n− 1 , m ) ⊆ N for some r 1 n− 1 ∈ R and m ∈ M , then either …

Strong metric dimension of the prime ideal sum graph of a commutative ring Open

Praveen Mathil, Jitender Kumar, R. Nikandish · 2023

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and on…

Metric dimension in a prime ideal sum graph of a commutative ring Open

M. Adlifard, Sh. Niknejad, R. Nikandish · 2023

The prime ideal sum graph of a commutative unital ring $R$, denoted by $PIS(R)$, is an undirect and simple graph whose vertices are non-trivial ideals of $R$ and there exists and edge between to distinct vertices if and only if their sum i…

Computing the strong metric dimension for co-maximal ideal graphs of commutative rings Open

R. Shahriyari, R. Nikandish, A. Tehranian, H. Rasouli · 2022

Let $R$ be a commutative ring with identity. The co-maximal ideal graph of $R$, denoted by $Γ(R)$, is a simple graph whose vertices are proper ideals of $R$ which are not contained in the Jacobson radical of $R$ and two distinct vertices $…

The weakly zero-divisor graph of a commutative ring Open

M. J. Nikmehr, Abdolreza Azadi, R. Nikandish · 2021

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R is the undirected (simple) graph W Γ(R) with vertex set Z(R) * , and two distinct vertices x and y are adjacent if…

On Perfect Co-Annihilating-Ideal Graph of a Commutative Artinian Ring Open

S. M. Saadat Mirghadim, M. J. Nikmehr, R. Nikandish · 2021

Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by AR, is a graph whose vertex set is the set of all non-zero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) ∩ Ann(…

On weakly $1$-absorbing prime ideals of commutative rings Open

M. J. Nikmehr, R. Nikandish, A. Yassine · 2021

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if f…

When the annihilator graph of a commutative ring is planar or toroidal? Open

M. Bakhtyiari, R. Nikandish, M. J. Nikmehr · 2020

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the undirected graph AG(R) with the vertex set Z(R)* = Z(R) \ {0}, and two distinct vertices x and y are adj…

On the metric dimension of strongly annihilating-ideal graphs of commutative rings Open

V. Soleymanivarniab, R. Nikandish, Abolfazl Tehranian · 2020

Some classes of perfect strongly annihilating-ideal graphs associated with commutative rings Open

Mitra Jalali, Abolfazl Tehranian, R. Nikandish, H. Rasouli · 2020

summary:Let $R$ be a commutative ring with identity and $A(R)$ be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of $R$ is defined as the graph ${\rm SAG}(R)$ with the vertex set $A(R)^*=A(R)\setminus\{0\…

A class of well-covered and vertex decomposable graphs arising from rings Open

Morteza Vafaei, Abolfazl Tehranian, R. Nikandish · 2020

Let $ mathbb {Z}_{n} $ be the ring of integers modulo $ n $. The unitary Cayley graph of $ mathbb {Z}_{n} $ is defined as the graph $ G( mathbb {Z}_{n} ) $ with the vertex set $ mathbb {Z}_{n} $ and two distinct vertices $a,b$ are adjacent…

When a total graph associated with a commutative ring is perfect? Open

Mitra Jalali, R. Nikandish, Abolfazl Tehranian · 2020

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The total graph of R is the graph T(?(R)) whose vertices are all elements of R, and two distinct vertices x and y are adjacent if and only if x + y ?…

On the planarity and perfectness of annihilator ideal graphs Open

R. Nikandish, M. J. Nikmehr, Mehdi Hosseini · 2019

Let $R$ be a commutative ring with unity. The annihilator ideal graph of $R$, denoted by $\Gamma _{\mathrm{Ann}} (R) $, is a graph whose vertices are all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and o…

Coloring of cozero-divisor graphs of commutative von Neumann regular rings Open

R. Nikandish, M. J. Nikmehr, M. Bakhtyiari · 2018

Let $R$ be a commutative ring with non-zero identity. The cozero-divisor graph of $R$, denoted by $Γ^{\prime}(R)$, is a graph with vertices in $W^*(R)$, which is the set of all non-zero and non-unit elements of $R$, and two distinct vertic…

The extended zero-divisor graph of a commutative ring II Open

M. Bakhtyiari, M. J. Nikmehr, R. Nikandish · 2017

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The extended zero-divisor graph of $R$ is the undirected (simple) graph $\\Gamma'(R)$ with the vertex set $Z(R)^*=Z(R)\\setminus\\{0\\}$, and t…

On the Structure of the Power Graph and the Enhanced Power Graph of a Group Open

Ghodratollah Aalipour, Saieed Akbari, Peter J‎. Cameron, R. Nikandish, Farzad Shaveisi · 2017

Let $G$ be a group‎. ‎The power graph of $G$ is a graph with the vertex‎ ‎set $G$‎, ‎having an edge between two elements whenever one is a power of the other‎. ‎We characterize nilpotent groups whose power graphs have finite independence n…

More on the annihilator graph of a commutative ring Open

M. J. Nikmehr, R. Nikandish, M. Bakhtyiari · 2017

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The annihilator graph of $R$ is defined as the undirected graph $AG(R)$ with the vertex set $Z(R)^*=Z(R)\\setminus\\{0\\}$, and two distinct ve…

When the Annihilator Graph of a Commutative Ring Is Planar or Toroidal? Open

M. J. Nikmehr, R. Nikandish, Moharam Bakhtyiari · 2017

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The annihilator graph of $R$ is defined as the undirected graph $AG(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct verti…

On the structure of compact graphs Open

R. Nikandish, Farzad Shaveisi · 2017

A simple graph $G$ is called a compact graph if $G$ contains no isolated vertices and for each pair $x$, $y$ of non-adjacent vertices of $G$, there is a vertex $z$ with $N(x)\cup N(y)\subseteq N(z)$, where $N(v)$ is the nei…

Some Properties of the Nil-Graphs of Ideals of Commutative Rings Open

R. Nikandish, Farzad Shaveisi · 2016

Let $R$ be a commutative ring with identity and ${\rm Nil}(R)$ be the set of nilpotent elements of $R$. The nil-graph of ideals of $R$ is defined as the graph $\mathbb{AG}_N(R)$ whose vertex set is $\{I:\ (0)\neq I\lhd R$ and there exists …

The annihilating-ideal graph of Z n is weakly perfect Open

R. Nikandish, Hamidreza Maimani, Hasan Izanloo · 2016

A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a commutative ring with identity and $\mathbb{A}(R)$ be the set of ideals with non-zero annihilator. The annihilating-ideal graph of $R$ i…

On the structure of the power graph and the enhanced power graph of a\n group Open

Ghodratollah Aalipour, Saieed Akbari, Peter J‎. Cameron, R. Nikandish, Farzad Shaveisi · 2016

Let $G$ be a group. The \\emph{power graph} of $G$ is a graph with the vertex\nset $G$, having an edge between two elements whenever one is a power of the\nother. We characterize nilpotent groups whose power graphs have finite\nindependenc…

THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING Open

Abolfazl Alibemani, M. Bakhtyiari, R. Nikandish, M. J. Nikmehr · 2015

Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by ${\Gamma}_{Ann}(R)$, is a graph whose vertices are all non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if $I{\cap}Ann(J…

Domination number in the annihilating-ideal graphs of commutative rings Open

R. Nikandish, Hamid Reza Maimani, Sima Kiani · 2015

Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(R)\{0} and two distinct vertices I and J are ad…

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