Journal ArticleDOI

# The total graph of a commutative ring

01 Oct 2008-Journal of Algebra (Academic Press)-Vol. 320, Iss: 7, pp 2706-2719
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.
##### Citations
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Book ChapterDOI
25 Sep 2007

425 citations

Book ChapterDOI
01 Jan 2011
TL;DR: A survey of the recent and active area of zero-divisor graphs of commutative rings can be found in this paper, followed by a historical overview and an extensive bibliography.
Abstract: This article surveys the recent and active area of zero-divisor graphs of commutative rings. Notable algebraic and graphical results are given, followed by a historical overview and an extensive bibliography.

153 citations

### Cites background from "The total graph of a commutative ri..."

• ...direction would be to start with a commutative ring R and use a different set of vertices or adjacency relation. For example, see [61, 75], or [ 11 ]....

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Journal ArticleDOI
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

Posted Content
TL;DR: In this article, it was shown that the annihilating-ideal graph of a commutative ring with a set of ideals with nonzero annihilator is a connected graph.
Abstract: Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the {\it annihilating-ideal graph} of $R$, denoted by ${\Bbb{AG}}(R)$. It is the (undirected) graph with vertices ${\Bbb{A}}(R)^*:={\Bbb{A}}(R)\setminus\{(0)\}$, and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. First, we study some finiteness conditions of ${\Bbb{AG}}(R)$. For instance, it is shown that if $R$ is not a domain, then ${\Bbb{AG}}(R)$ has ACC (resp., DCC) on vertices if and only if $R$ is Noetherian (resp., Artinian). Moreover, the set of vertices of ${\Bbb{AG}}(R)$ 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 $R$, ${\Bbb{AG}}(R)$ has $n$ vertices $(n\geq 1)$ if and only if $R$ has only $n$ nonzero proper ideals. Next, we study the connectivity of ${\Bbb{AG}}(R)$. It is shown that ${\Bbb{AG}}(R)$ is a connected graph and $diam(\Bbb{AG})(R)\leq 3$ and if ${\Bbb{AG}}(R)$ contains a cycle, then $gr({\Bbb{AG}}(R))\leq 4$. Also, rings $R$ for which the graph ${\Bbb{AG}}(R)$ is complete or star, are characterized, as well as rings $R$ for which every vertex of ${\Bbb{AG}}(R)$ is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.

127 citations

### Cites background from "The total graph of a commutative ri..."

• ...Recently Anderson and Badawi in [2] have introduced and investigated the total graph of R, denoted by T (Γ(R))....

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• ...In the literature, there are many papers on assigning a graph to a ring, a group, semigroup or a module (see, for example, [1–11, 14–18])....

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Journal ArticleDOI
TL;DR: In this article, it was shown that if R is a commutative ring and S is a finite ring, then T ( Γ ( R ) is a Hamiltonian graph.

117 citations

### Cites background from "The total graph of a commutative ri..."

• ...The total graph of R denoted by T (Γ (R)) was introduced in [8], as the graph with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x + y ∈ Z(R)....

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• ...Anderson and Badawi showed that if R is a commutative ring, Z(R) is not an ideal and T (Γ (R)) is connected, then diam(Reg(Γ (R))) 6 diam(T (Γ (R))), see [8]....

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• ...In [8] it was proved that if R is a commutative ring such that T (Γ (R)) is connected and Z(R) is not an ideal, then diam(T (Γ (R))) 6 diam(Reg(Γ (R)))+ 2....

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• ...In [8] it was proved that for every commutative ring R if Z(R) is not an ideal of R, then T (Γ (R)) is connected if and only if the ideal generated by Z(R) is R (i....

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Book
01 Jan 1969

16,023 citations

### "The total graph of a commutative ri..." refers background in this paper

• ...A general reference for graph theory is [8]....

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Journal ArticleDOI
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

### "The total graph of a commutative ri..." refers background in this paper

• ...In [3], Anderson and Livingston introduced the zero-divisor graph of R, denoted by Γ (R), as the (undirected) graph with vertices Z(R)∗ = Z(R)\{0}, the set of nonzero zero-divisors of R, and for distinct x, y ∈ Z(R)∗, the vertices x and y are adjacent if and only if xy = 0....

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Journal ArticleDOI
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

### "The total graph of a commutative ri..." refers background in this paper

• ...This concept is due to Beck [7], who let all the elements of R be vertices and was mainly interested in colorings....

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Book
01 Jan 1979
TL;DR: In an elementary text book, the reader gains an overall understanding of well-known standard results, and yet at the same time constant hints of, and guidelines into, the higher levels of the subject.
Abstract: From the reviews: "Bla Bollob's introductory course on graph theory deserves to be considered as a watershed in the development of this theory as a serious academic subject...The book has chapters on electrical networks, flows, connectivity and matchings, extremal problems, colouring, Ramsey theory, random graphs, and graphs and groups. Each chapter starts at a measured and gentle pace. Classical results are proved and new insight is provided, with the examples at the end of each chapter fully supplementing the text...Even so this allows an introduction not only to some of the deeper results but, more vitally, provides outlines of, and firm insights into, their proofs. Thus in an elementary text book, we gain an overall understanding of well-known standard results, and yet at the same time constant hints of, and guidelines into, the higher levels of the subject. It is this aspect of the book which should guarantee it a permanent place in the literature."

603 citations

Book
21 Mar 1988

574 citations

### "The total graph of a commutative ri..." refers background in this paper

• ...A standard notation for this “idealized ring” is R(+)M ; see [10] for basic properties of rings resulting from the idealization construction....

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• ...General references for ring theory are [10] and [11]....

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