The prime ideals intersection graph of a ring

M. J. Nikmehr; B. Soleymanzadeh

Displaying similar documents to “The prime ideals intersection graph of a ring”

Comaximal graph of $C (X)$

Mehdi Badie (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this article we study the comaximal graph $Γ_{_{2}}^{'} C (X)$ of the ring $C (X)$ . We have tried to associate the graph properties of $Γ_{_{2}}^{'} C (X)$ , the ring properties of $C (X)$ and the topological properties of $X$ . Radius, girth, dominating number and clique number of the $Γ_{_{2}}^{'} C (X)$ are investigated. We have shown that $2 \leq Rad Γ_{_{2}}^{'} C (X) \leq 3$ and if $| X | > 2$ then $girth Γ_{_{2}}^{'} C (X) = 3$ . We give some topological properties of $X$ equivalent to graph properties of $Γ_{_{2}}^{'} C (X)$ . Finally we have proved that $X$ is an almost $P$ -space which does not have isolated points if and only if $C (X)$ is an almost...

On the recognizability of some projective general linear groups by the prime graph

Masoumeh Sajjadi (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group. The prime graph of $G$ is a simple graph $Γ (G)$ whose vertex set is $π (G)$ and two distinct vertices $p$ and $q$ are joined by an edge if and only if $G$ has an element of order $p q$ . A group $G$ is called $k$ -recognizable by prime graph if there exist exactly $k$ nonisomorphic groups $H$ satisfying the condition $Γ (G) = Γ (H)$ . A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that $PGL (2, p^{α})$ is recognizable, if $p$ is an odd prime and $α > 1$ is odd. But for even $α$ , only...

On graph associated to co-ideals of commutative semirings

Yahya Talebi, Atefeh Darzi (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a commutative semiring with non-zero identity. In this paper, we introduce and study the graph $Ω (R)$ whose vertices are all elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if the product of the co-ideals generated by $x$ and $y$ is $R$ . Also, we study the interplay between the graph-theoretic properties of this graph and some algebraic properties of semirings. Finally, we present some relationships between the zero-divisor graph $Γ (R)$ and $Ω (R)$ .

Some results on the co-intersection graph of submodules of a module

Lotf Ali Mahdavi, Yahya Talebi (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a ring with identity and $M$ be a unitary left $R$ -module. The co-intersection graph of proper submodules of $M$ , denoted by $Ω (M)$ , is an undirected simple graph whose vertex set $V (Ω)$ is a set of all nontrivial submodules of $M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $N + K \neq M$ . We study the connectivity, the core and the clique number of $Ω (M)$ . Also, we provide some conditions on the module $M$ , under which the clique number of $Ω (M)$ is infinite and $Ω (M)$ is a planar graph. Moreover, we give...

Symmetric and reversible properties of bi-amalgamated rings

Antonysamy Aruldoss, Chelliah Selvaraj (2024)

Czechoslovak Mathematical Journal

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Let $f : A \to B$ and $g : A \to C$ be two ring homomorphisms and let $K$ and $K^{'}$ be two ideals of $B$ and $C$ , respectively, such that $f^{- 1} (K) = g^{- 1} (K^{'})$ . We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring $A ⋈^{f, g} (K, K^{'})$ of $A$ with $(B, C)$ along $(K, K^{'})$ with respect to $(f, g)$ .

Classification of rings with toroidal Jacobson graph

Krishnan Selvakumar, Manoharan Subajini (2016)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with nonzero identity and $J (R)$ the Jacobson radical of $R$ . The Jacobson graph of $R$ , denoted by $𝔍_{R}$ , is defined as the graph with vertex set $R ∖ J (R)$ such that two distinct vertices $x$ and $y$ are adjacent if and only if $1 - x y$ is not a unit of $R$ . The genus of a simple graph $G$ is the smallest nonnegative integer $n$ such that $G$ can be embedded into an orientable surface $S_{n}$ . In this paper, we investigate the genus number of the compact Riemann surface in which $𝔍_{R}$ can be embedded and...

The Turán number of the graph $3 P_{4}$

Halina Bielak, Sebastian Kieliszek (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $e x (n, G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_{i}$ denote a path consisting of $i$ vertices and let $m P_{i}$ denote $m$ disjoint copies of $P_{i}$ . In this paper we count $e x (n, 3 P_{4})$ .

Gaussian and Prüfer conditions in bi-amalgamated algebras

Najib Mahdou, Moutu Abdou Salam Moutui (2020)

Czechoslovak Mathematical Journal

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Let $f : A \to B$ and $g : A \to C$ be two ring homomorphisms and let $J$ and $J^{'}$ be ideals of $B$ and $C$ , respectively, such that $f^{- 1} (J) = g^{- 1} (J^{'})$ . In this paper, we investigate the transfer of the notions of Gaussian and Prüfer rings to the bi-amalgamation of $A$ with $(B, C)$ along $(J, J^{'})$ with respect to $(f, g)$ (denoted by $A ⋈^{f, g} (J, J^{'})),$ introduced and studied by S. Kabbaj, K. Louartiti and M. Tamekkante in 2013. Our results recover well known results on amalgamations in C. A. Finocchiaro (2014) and generate new original examples of rings possessing these properties. ...

A note on solvable vertex stabilizers of $s$ -transitive graphs of prime valency

Song-Tao Guo, Hailong Hou, Yong Xu (2015)

Czechoslovak Mathematical Journal

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A graph $X$ , with a group $G$ of automorphisms of $X$ , is said to be $(G, s)$ -transitive, for some $s \geq 1$ , if $G$ is transitive on $s$ -arcs but not on $(s + 1)$ -arcs. Let $X$ be a connected $(G, s)$ -transitive graph of prime valency $p \geq 5$ , and $G_{v}$ the vertex stabilizer of a vertex $v \in V (X)$ . Suppose that $G_{v}$ is solvable. Weiss (1974) proved that $| G_{v} {| ∣ p (p - 1)}^{2}$ . In this paper, we prove that $G_{v} ≅ (ℤ_{p} ⋊ ℤ_{m}) \times ℤ_{n}$ for some positive integers $m$ and $n$ such that $n div m$ and $m ∣ p - 1$ .

Some results on the annihilator graph of a commutative ring

Mojgan Afkhami, Kazem Khashyarmanesh, Zohreh Rajabi (2017)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring. The annihilator graph of $R$ , denoted by $AG (R)$ , is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${ann}_{R} (x y) \neq {ann}_{R} (x) \cup {ann}_{R} (y)$ , where for $z \in R$ , ${ann}_{R} (z) = {r \in R : r z = 0}$ . In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$ , $2$ or $3$ . Also, we investigate some properties...

Annihilators of skew derivations with Engel conditions on prime rings

Taylan Pehlivan, Emine Albas (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring $Q$ , $C$ the extended centroid of $R$ and $a \in R$ . Suppose that $δ$ is a nonzero $σ$ -derivation of $R$ such that $a {[δ (x^{n}), x^{n}]}_{k} = 0$ for all $x \in R$ , where $σ$ is an automorphism of $R$ , $n$ and $k$ are fixed positive integers. Then $a = 0$ .

The small Ree group $^{2} G_{2} (3^{2 n + 1})$ and related graph

Alireza K. Asboei, Seyed S. S. Amiri (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group. The main supergraph $𝒮 (G)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and only if $o (x) ∣ o (y)$ or $o (y) ∣ o (x)$ . In this paper, we will show that $G ≅^{2} G_{2} (3^{2 n + 1})$ if and only if $𝒮 (G) ≅ 𝒮 (^{2} G_{2} (3^{2 n + 1}))$ . As a main consequence of our result we conclude that Thompson’s problem is true for the small Ree group $^{2} G_{2} (3^{2 n + 1})$ .

Even factor of bridgeless graphs containing two specified edges

Nastaran Haghparast, Dariush Kiani (2018)

Czechoslovak Mathematical Journal

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An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let $G$ be a bridgeless simple graph with minimum degree at least $3$ . Jackson and Yoshimoto (2007) showed that $G$ has an even factor containing two arbitrary prescribed edges. They also proved that $G$ has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges $e_{1}$ and $e_{2}$ of $G$ , there is an even factor containing $e_{1}$ and $e_{2}$ ...