Annihilating-ideal Graph Research Articles

The undirected annihilating-ideal graphs over non-commutative rings

The undirected annihilating-ideal graphs over non-commutative rings

Metric and Upper Dimension of Extended Annihilating-Ideal Graphs

The metric dimension problem is called navigation problem due to its application to robot navigation in space. Further this concept has wide applications in motion planning, sonar and loran station, and so on. In this paper, we study certain results on the metric dimension, upper dimension and resolving number of extended annihilating-ideal graph [Formula: see text] associated to a commutative ring [Formula: see text], denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. Here we prove the finiteness conditions of [Formula: see text] and [Formula: see text]. In addition, we characterize [Formula: see text], [Formula: see text] and [Formula: see text] for artinian rings and the direct product of rings.

Some remarks on the dominating sets of the annihilating-ideal graph of a commutative ring

Some remarks on the dominating sets of the annihilating-ideal graph of a commutative ring

Coloring in essential annihilating-ideal graphs of commutative rings

Coloring in essential annihilating-ideal graphs of commutative rings

On two extensions of the annihilating-ideal graph of commutative rings

Abstract Let R be a commutative ring with A ⁢ ( R ) {A(R)} its set of annihilating-ideals. The extended annihilating-ideal graph of R, denoted by AG ¯ ⁢ ( R ) {\overline{\mathrm{AG}}(R)} , is an undirected graph with vertex set A ⁢ ( R ) * = A ⁢ ( R ) ∖ { 0 } {A(R)^{*}=A(R)\setminus\{0\}} and two vertices I 1 {I_{1}} and I 2 {I_{2}} are adjacent if and only if I 1 m ⁢ I 2 n = 0 {I_{1}^{m}I_{2}^{n}=0} with I 1 m ≠ 0 {I_{1}^{m}\neq 0} and I 2 n ≠ 0 {I_{2}^{n}\neq 0} , for some positive integers m and n. In this paper, we first study some basic properties of AG ¯ ⁢ ( R ) {\overline{\mathrm{AG}}(R)} and then we investigate the relationship between the extended annihilating-ideal graph AG ¯ ⁢ ( R ) {\overline{\mathrm{AG}}(R)} , the annihilator-ideal graph A I ⁢ ( R ) {A_{I}(R)} and the annihilating-ideal graph AG ⁢ ( R ) {\mathrm{AG}(R)} of a commutative ring R.

On the genus of strong annihilating-ideal graph of commutative rings

Let [Formula: see text] be a commutative ring with unity and [Formula: see text] be the set of annihilating-ideals of [Formula: see text]. The strong annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph with vertex set [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. In this paper, first we characterize commutative Artinian rings whose strong annihilating-ideal graph is isomorphic to some well-known graphs and then we classify commutative Artinian rings whose strong annihilating-ideal graph is planar, toroidal or projective.

Computing the (forcing) strong metric dimension in strongly annihilating-ideal graphs

Computing the (forcing) strong metric dimension in strongly annihilating-ideal graphs

The weakly annihilating-ideal graph of a commutative ring

Let R be a commutative ring with non-zero identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r∈R\ 0 such that Ir=(0). Let A(R) denote the set of all annihilating ideals of R and A(R)∗=A(R)\{(0)}. In this article, we introduce and investigate the weakly annihilating-ideal graph of R denoted by WAG(R). It is the undirected graph whose vertex set is A(R)∗ and two distinct vertices I, J are adjacent in this graph if and only if there exist non-zero ideals A, B of R with A ⊆ ann(I) and B ⊆ ann(J) such that AB=(0). The aim of this article is to study the interplay between the ring-theoretic properties of R and the graph-theoretic properties of WAG(R). We discuss some results regarding the connectedness of WAG(R) and determine its diameter and girth. Moreover, we provide some conditions under which WAG(R) and AG(R) are identical.

Class of Crosscap Two Graphs Arising from Lattices–I

Let L be a lattice. The annihilating-ideal graph of L is a simple graph whose vertex set is the set of all nontrivial ideals of L and whose two distinct vertices I and J are adjacent if and only if I∧J=0. In this paper, crosscap two annihilating-ideal graphs of lattices with at most four atoms are characterized. These characterizations provide the classes of multipartite graphs, which are embedded in the Klein bottle.

Class of crosscap two graphs arising from lattices-Ⅱ

<abstract><p>In this series of papers, we study the crosscap two embedding of a class of multipartite graphs, namely, annihilating-ideal graphs of a lattice. In Part 1 of the series [Class of crosscap two graphs arising from lattices-Ⅰ, <italic>Mathematics</italic>, <bold>11</bold> (2023), 1-26], we classified lattices with the number of atoms less than or equal to 4, whose annihilating-ideal graph can be embedded in the Klein bottle. In this paper, which is Part 2 of the series, we classify all finite lattices with at least 5 atoms whose annihilating-ideal graph is embedded in crosscap two surfaces. These characterizations help us to identify classes of multipartite graphs, which are embedded in the Klein bottle.</p></abstract>

Strong metric dimension in annihilating-ideal graph of commutative rings

Strong metric dimension in annihilating-ideal graph of commutative rings

On the essential annihilating-ideal graph of commutative rings

Let R be a commutative ring with unity, A(R) be the set of all annihilating-ideals of R and A*(R) = A(R) \ {0}. In this paper, we introduced and studied the essential annihilating-ideal graph of R, denoted by EG(R), with vertex set A*(R) and two distinct vertices I1 and I2 are adjacent if and only if Ann(I1I2) is an essential ideal of R. We prove that EG(R) is a connected graph with diameter at most three and girth at most four if EG(R) contains a cycle. Furthermore, the rings R are characterized for which EG(R) is a star or a complete graph. Finally, we classify all the Artinian rings R for which EG(R) is isomorphic to some well-known graphs.

On a spanning subgraph of the annihilating-ideal graph of a commutative ring

The rings considered in this paper are commutative with identity which are not integral domains. Let [Formula: see text] be a ring. Let us denote the set of all annihilating ideals of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. With [Formula: see text], we associate an undirected graph, denoted by [Formula: see text], whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent in this graph if and only if [Formula: see text] and [Formula: see text]. The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-theoretic properties of [Formula: see text].

Extended annihilating-ideal graph of a commutative ring

Extended annihilating-ideal graph of a commutative ring

ANNIHILATING-IDEAL GRAPH OF C(X)

In this article the annihilating-ideal graph of the ring C(X) is studied. We have tried to associate the graph properties of AG(X), the ring properties of C(X) and the topological properties of X. It is shown that X has an isolated point if and only if R is a direct summand of C(X) and this happens if and only if AG(X) is not triangulated. Radius, girth, dominating number and clique number of the AG(X) are investigated. It is proved that c(X) <= dt(AG(X)) ,= w(X) and wAG(X) = χAG(X) = c(X).

DOMINATION NUMBER IN THE ANNIHILATING-SUBMODULE GRAPH OF MODULES OVER COMMUTATIVE RINGS

Let $M$ be a module over a commutative ring $R$. The&#x0D; annihilating-submodule graph of $M$, denoted by $AG(M)$, is a&#x0D; simple undirected graph in which a non-zero submodule $N$ of $M$&#x0D; is a vertex if and only if there exists a non-zero proper&#x0D; submodule $K$ of $M$ such that $NK=(0)$, where $NK$, the product&#x0D; of $N$ and $K$, is denoted by $(N:M)(K:M)M$ and two distinct&#x0D; vertices $N$ and $K$ are adjacent if and only if $NK=(0)$. This&#x0D; graph is a submodule version of the annihilating-ideal graph and&#x0D; under some conditions, is isomorphic with an induced subgraph of&#x0D; the Zariski topology-graph $G(\tau_T)$ which was introduced in [H.&#x0D; Ansari-Toroghy and S. Habibi, Comm. Algebra, 42(2014), 3283-3296].&#x0D; In this paper, we study the domination number of $AG(M)$ and some&#x0D; connections between the graph-theoretic properties of $AG(M)$ and&#x0D; algebraic properties of module $M$.

Notes on the zero-divisor graph and annihilating-ideal graph of a reduced ring

Abstract We translate some graph properties of 𝔸𝔾(R) and Γ(R) to some topological properties of Zariski topology. We prove that the facts “(1) The zero ideal of R is an anti fixed-place ideal. (2) Min(R) does not have any isolated point. (3) Rad(𝔸𝔾 (R)) = 3. (4) Rad(Γ(R)) = 3. (5) Γ(R) is triangulated (6) 𝔸𝔾 (R) is triangulated.” are equivalent. Also, we show that if the zero ideal of a ring R is a fixed-place ideal, then dt t (𝔸𝔾 (R)) = |ℬ(R)| and also if in addition |Min(R)| &gt; 2, then dt(𝔸𝔾 (R)) = |ℬ (R)|. Finally, it is shown that dt(𝔸𝔾 (R)) is finite if and only if dt t (𝔸𝔾 (R)) is finite if and only if Min(R) is finite.

The exact annihilating-ideal graph of a commutative ring

The rings considered in this article are commutative with identity. For an ideal $I$ of a ring $R$, we denote the annihilator of $I$ in $R$ by $Ann(I)$. An ideal $I$ of a ring $R$ is said to be an exact annihilating ideal if there exists a non-zero ideal $J$ of $R$ such that $Ann(I) = J$ and $Ann(J) = I$. For a ring $R$, we denote the set of all exact annihilating ideals of $R$ by $\mathbb{EA}(R)$ and $\mathbb{EA}(R)\backslash \{(0)\}$ by $\mathbb{EA}(R)^{*}$. Let $R$ be a ring such that $\mathbb{EA}(R)^{*}\neq \emptyset$. With $R$, in [Exact Annihilating-ideal graph of commutative rings, {\it J. Algebra and Related Topics} {\bf 5}(1) (2017) 27-33] P.T. Lalchandani introduced and investigated an undirected graph called the exact annihilating-ideal graph of $R$, denoted by $\mathbb{EAG}(R)$ whose vertex set is $\mathbb{EA}(R)^{*}$ and distinct vertices $I$ and $J$ are adjacent if and only if $Ann(I) = J$ and $Ann(J) = I$. In this article, we continue the study of the exact annihilating-ideal graph of a ring. In Section 2 , we prove some basic properties of exact annihilating ideals of a commutative ring and we provide several examples. In Section 3, we determine the structure of $\mathbb{EAG}(R)$, where either $R$ is a special principal ideal ring or $R$ is a reduced ring which admits only a finite number of minimal prime ideals.

ON WEAKLY PERFECT ANNIHILATING-IDEAL GRAPHS

AbstractWe prove that the annihilating-ideal graph of a commutative semigroup with unity is, in general, not weakly perfect. This settles the conjecture of DeMeyer and Schneider [‘The annihilating-ideal graph of commutative semigroups’, J. Algebra469 (2017), 402–420]. Further, we prove that the zero-divisor graphs of semigroups with respect to semiprime ideals are weakly perfect. This enables us to produce a large class of examples of weakly perfect zero-divisor graphs from a fixed semigroup by choosing different semiprime ideals.

THE METRIC DIMENSION OF THE ANNIHILATING-IDEAL GRAPH OF A FINITE COMMUTATIVE RING

AbstractWe determine the metric dimension of the annihilating-ideal graph of a local finite commutative principal ring and a finite commutative principal ring with two maximal ideals. We also find bounds for the metric dimension of the annihilating-ideal graph of an arbitrary finite commutative principal ring.