# On the genus of the graph associated to a commutative ring

02 Nov 2017-Discrete Mathematics, Algorithms and Applications (World Scientific Publishing Company)-Vol. 09, Iss: 05, pp 1750058

TL;DR: The commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one is characterized, whose vertex set is the set of all non-trivial ideals of R.

Abstract: Let R be a commutative ring with identity. We consider a simple graph associated with R, denoted by ΩR∗, whose vertex set is the set of all non-trivial ideals of R and two distinct vertices I and J are adjacent whenever JAnn(I) = (0) or IAnn(J) = (0). In this paper, we characterize the commutative Artinian non-local ring R for which ΩR∗ has genus one and crosscap one.

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01 Dec 2018TL;DR: In this paper, all commutative Artinian nonlocal rings with genus one have genus one and the annihilating-ideal graph is defined as the graph with the vertices of two distinct vertices that are adjacent if and only if their genus is genus one.

Abstract: Let $R$ be a non-domain commutative ring with identity and $A^*(R)$ be theset of non-zero ideals with non-zero annihilators. We call an ideal $I$ of $R$, anannihilating-ideal if there exists a non-zero ideal $J$ of $R$ such that $IJ = (0)$.The annihilating-ideal graph of $R$ is defined as the graph $AG(R)$ with the vertexset $A^*(R)$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ =(0)$. In this paper, we characterize all commutative Artinian nonlocal rings $R$for which $AG(R)$ has genus one.

3 citations

### Cites background from "On the genus of the graph associate..."

...[20] If (R,m) is a local ring and there is an ideal I of R such that I 6= mi for every i, then R has at least three distinct non-trivial ideals J, K and L such that J, K, L 6= mi for every i....

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TL;DR: In this paper, the authors characterize all Artinian commutative nonlocal rings R with genus one k-annihilating-ideal hypergraphs and show that all of them have genus one vertices.

Abstract: Let R be a commutative ring and k an integer greater than 2 and let $${\cal A}(R, k)$$
be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph of R, denoted by $${\cal A}{{\cal G}_k}(R)$$
, is a hypergraph with vertex set $${\cal A}(R, k)$$
, and for distinct elements I1, …, Ik in $${\cal A}(R, k)$$
, the set {I1, I2, …, Ik} is an edge of $${\cal A}{{\cal G}_k}(R)$$
if and only if $$\prod\limits_{i = 1}^k {{I_i} = (0)} $$
and the product of any (k − 1) elements of the set {I1, I2,…, Ik} is nonzero. In this paper, we characterize all Artinian commutative nonlocal rings R whose $${\cal A}{{\cal G}_3}(R)$$
has genus one.

3 citations

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01 Dec 2021TL;DR: The annihilator-inclusion ideal graph of R, denoted by ξR, is a graph whose vertex set is the of all non-zero proper ideals of R, and two distinct vertices $I$ and $J$ are adjacent if and only if either ANN(I) ⊆ J or ANN(J)⊆ I.

Abstract: Let R be a commutative ring with non-zero identity. The annihilator-inclusion ideal graph of R , denoted by ξR, is a graph whose vertex set is the of allnon-zero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacentif and only if either Ann(I) ⊆ J or Ann(J) ⊆ I. In this paper, we investigate the basicproperties of the graph ξR. In particular, we showthat ξR is a connected graph with diameter at most three, andhas girth 3 or ∞. Furthermore, we determine all isomorphic classes of non-local Artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.

2 citations

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TL;DR: This paper characterize all Artinian rings [Formula: see text] for which the genus of [Form formula]: see text is zero or one.

Abstract: 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 nonzero proper ideals of R and two distinct vertices I ...

1 citations

01 Jan 2018

TL;DR: It is shown that ΩR∗ is a connected graph with diam(ΩR ∼3) ≤ 3 unless R is isomorphic to a direct product of two fields and all commutative rings R with at least two maximal ideals for which Ω R∗ are planar are characterized.

Abstract: In this paper some properties of the complement of a new graph associated with a commutative ring are investigated ....

##### References

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01 Jan 1969

TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:

Abstract: * Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory

4,168 citations

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TL;DR: For each commutative ring R we associate a simple graph Γ(R) as discussed by the authors, and we investigate the interplay between the ring-theoretic properties of R and the graph-theory properties of Γ (R).

1,087 citations

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TL;DR: In this article, the authors present the idea of coloring of a commutative ring and show that the existence of an infinite clique implies that the clique R = co implies that there exists an infinitely many cliques.

956 citations

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TL;DR: In this article, the authors introduced and investigated the total graph of R, denoted by T ( Γ ( R ) ), which is the (undirected) graph with all elements of R as vertices.

290 citations

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TL;DR: In this paper, it was shown that if R is not a domain, then 𝔸&#x 1d53e;(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if r is Noetherian or a decomposable ring.

Abstract: Let R be a commutative ring, with 𝔸(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). It is the (undirected) graph with vertices 𝔸(R)* ≔ 𝔸(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of 𝔸𝔾(R). For instance, it is shown that if R is not a domain, then 𝔸𝔾(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of 𝔸𝔾(R) and the set of nonzero proper ideals of R have the same cardinality when R is either an Artinian or a decomposable ring. This yields for a ring R, 𝔸𝔾(R) has n vertices (n ≥ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of 𝔸𝔾(R). It is shown that 𝔸𝔾(R) is a connected graph and diam(𝔸𝔾)(R) ≤ 3 and if 𝔸𝔾(R) contains a cycle, then gr(𝔸𝔾(R)) ≤ 4. Also, rings R for which the graph 𝔸𝔾(R) is complete or star, are characterized, as well as rings R for which every vertex of 𝔸𝔾(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

129 citations