Some results on the annihilator graph of a commutative ring

Mojgan Afkhami; Kazem Khashyarmanesh; Zohreh Rajabi

Displaying similar documents to “Some results on the annihilator graph of a commutative ring”

Classification of rings with toroidal Jacobson graph

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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...

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

Mitra Jalali, Abolfazl Tehranian, Reza Nikandish, Hamid Rasouli (2020)

Commentationes Mathematicae Universitatis Carolinae

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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 $SAG (R)$ with the vertex set $A {(R)}^{*} = A (R) ∖ {0}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I \cap Ann (J) \neq (0)$ and $J \cap Ann (I) \neq (0)$ . In this paper, the perfectness of $SAG (R)$ for some classes of rings $R$ is investigated.

The prime ideals intersection graph of a ring

M. J. Nikmehr, B. Soleymanzadeh (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let $R$ be a commutative ring with unity and $U (R)$ be the set of unit elements of $R$ . In this paper, we introduce and investigate some properties of a new kind of graph on the ring $R$ , namely, the prime ideals intersection graph of $R$ , denoted by $G_{p} (R)$ . The $G_{p} (R)$ is a graph with vertex set $R^{*} - U (R)$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists a prime ideal $𝔭$ of $R$ such that $a, b \in 𝔭$ . We obtain necessary and sufficient conditions on $R$ such that $G_{p} (R)$ is disconnected. We find the diameter and...

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...

(Generalized) filter properties of the amalgamated algebra

Yusof Azimi (2022)

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Let $R$ and $S$ be commutative rings with unity, $f : R \to S$ a ring homomorphism and $J$ an ideal of $S$ . Then the subring $R ⋈^{f} J : = {(a, f (a) + j) ∣ a \in R$ and $j \in J}$ of $R \times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$ . In this paper, we determine when $R ⋈^{f} J$ is a (generalized) filter ring.

Degree sums of adjacent vertices for traceability of claw-free graphs

Tao Tian, Liming Xiong, Zhi-Hong Chen, Shipeng Wang (2022)

Czechoslovak Mathematical Journal

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The line graph of a graph $G$ , denoted by $L (G)$ , has $E (G)$ as its vertex set, where two vertices in $L (G)$ are adjacent if and only if the corresponding edges in $G$ have a vertex in common. For a graph $H$ , define ${\bar{σ}}_{2} (H) = min {d (u) + d (v) : u v \in E (H)}$ . Let $H$ be a 2-connected claw-free simple graph of order $n$ with $δ (H) \geq 3$ . We show that, if ${\bar{σ}}_{2} (H) \geq \frac{1}{7} (2 n - 5)$ and $n$ is sufficiently large, then either $H$ is traceable or the Ryjáček’s closure $cl (H) = L (G)$ , where $G$ is an essentially $2$ -edge-connected triangle-free graph that can be contracted to one of the two graphs of order 10...

Saturation numbers for linear forests $P_{6} + t P_{2}$

Jingru Yan (2023)

Czechoslovak Mathematical Journal

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A graph $G$ is $H$ -saturated if it contains no $H$ as a subgraph, but does contain $H$ after the addition of any edge in the complement of $G$ . The saturation number, $sat (n, H)$ , is the minimum number of edges of a graph in the set of all $H$ -saturated graphs of order $n$ . We determine the saturation number $sat (n, P_{6} + t P_{2})$ for $n \geq \frac{10}{3} t + 10$ and characterize the extremal graphs for $n > \frac{10}{3} t + 20$ .

P-injective group rings

Liang Shen (2020)

Czechoslovak Mathematical Journal

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A ring $R$ is called right P-injective if every homomorphism from a principal right ideal of $R$ to $R_{R}$ can be extended to a homomorphism from $R_{R}$ to $R_{R}$ . Let $R$ be a ring and $G$ a group. Based on a result of Nicholson and Yousif, we prove that the group ring $RG$ is right P-injective if and only if (a) $R$ is right P-injective; (b) $G$ is locally finite; and (c) for any finite subgroup $H$ of $G$ and any principal right ideal $I$ of $RH$ , if $f \in {Hom}_{R} (I_{R}, R_{R})$ , then there exists $g \in {Hom}_{R} ({RH}_{R}, R_{R})$ such that ${g |}_{I} = f$ . Similarly, we also obtain equivalent...

Semicommutativity of the rings relative to prime radical

Handan Kose, Burcu Ungor (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$ -semicommutative. We prove that a ring $R$ is $P$ -semicommutative if and only if $R [x]$ is $P$ -semicommutative if and only if $R [x, x^{- 1}]$ is $P$ -semicommutative. Also, if $R [[x]]$ is $P$ -semicommutative, then $R$ is $P$ -semicommutative. The converse holds provided that $P (R)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$ , $R$ is $P$ -semicommutative...

Strongly 2-nil-clean rings with involutions

Huanyin Chen, Marjan Sheibani Abdolyousefi (2019)

Czechoslovak Mathematical Journal

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A $*$ -ring $R$ is strongly 2-nil- $*$ -clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$ -rings are obtained. We prove that a $*$ -ring $R$ is strongly 2-nil- $*$ -clean if and only if for all $a \in R$ , $a^{2} \in R$ is strongly nil- $*$ -clean, if and only if for any $a \in R$ there exists a $*$ -tripotent $e \in R$ such that $a - e \in R$ is nilpotent and $e a = a e$ , if and only if $R$ is a strongly $*$ -clean SN ring, if and only if $R$ is abelian, $J (R)$ is nil and $R / J (R)$ is $*$ -tripotent. Furthermore, we explore...

The $\circ$ operation and $*$ operation of Cohen-Macaulay bipartite graphs

Yulong Yang, Guangjun Zhu, Yijun Cui, Shiya Duan (2024)

Czechoslovak Mathematical Journal

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Let $G$ be a finite simple graph with the vertex set $V$ and let $I_{G}$ be its edge ideal in the polynomial ring $S = 𝕂 [V]$ . We compute the depth and the Castelnuovo-Mumford regularity of $S / I_{G}$ when $G = G_{1} \circ G_{2}$ or $G = G_{1} * G_{2}$ is a graph obtained from Cohen-Macaulay bipartite graphs $G_{1}$ , $G_{2}$ by the $\circ$ operation or $*$ operation, respectively.