# The essential ideal graph of a commutative ring

23 May 2017-Asian-european Journal of Mathematics (World Scientific Publishing Company)-Vol. 11, Iss: 04, pp 1850058

TL;DR: The study of the essential ideal graph of a commutative ring with identity was initiated in this article, where the authors investigated its properties and showed that it is a graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent whenever I + J is an essential ideal.

Abstract: Let R be a commutative ring with identity The essential ideal graph of R, denoted by ℰR, is a graph whose vertex set is the set of all nonzero proper ideals of R and two vertices I and J are adjacent whenever I + J is an essential ideal In this paper, we initiate the study of the essential ideal graph of a commutative ring and we investigate its properties

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TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.

Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

1,479 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 initiates the study of the co-annihilating graph of a commutative ring and investigates its properties.

Abstract: Let R be a commutative ring with identity and 𝔘R be the set of all non-zero non-units of R. The co-annihilating graph of R, denoted by 𝒞𝒜R, is a graph with vertex set 𝔘R and two vertices a and b are adjacent whenever Ann(a) ∩Ann(b) = (0). In this paper, we initiate the study of the co-annihilating graph of a commutative ring and we investigate its properties.

2 citations

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TL;DR: The sum-essential graph as mentioned in this paper is a graph whose vertices are all nontrivial submodules of a left R-module M and two distinct submodules are adjacent if and only if their sum is an essential submodule.

Abstract: The sum-essential graph 𝒮R(M) of a left R-module M is a graph whose vertices are all nontrivial submodules of M and two distinct submodules are adjacent if and only if their sum is an essential sub...

2 citations

### Cites methods from "The essential ideal graph of a comm..."

...lso study the proper sum-essential graph of M which is a subgraph of S R(M) generated by vertices which, as submodules of M are not essential. The graph S R(M) was introduced and studied by Amjadi in [5] in a very special case when R is a commutative ring and M = RR. There are other graphs relating submodule structure of a given module. For example comaximal left ideal graphs, i.e. graphs whose verti...

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26 Mar 2021

TL;DR: In this paper, the domination number of the essential ideal graph of a commutative ring with nonzero unity has been studied and a characterization for reduced rings with domination number 1 is given.

Abstract: Let A be a commutative ring with nonzero unity. The essential ideal graph of A, ℰA, is a graph with set of all nonzero proper ideals of A as the vertex set and two vertices I and J are adjacent whenever I+J is an essential ideal. In this article, we discuss about the domination number of the essential ideal graph of a commutative ring. We obtain a characterization for reduced rings to have the domination number 1. Also, we determine the domination parameters of essential ideal graph of the commutative rings ℤn and F1 × F2 × ··· × Fn, n ≥ 2.

##### References

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TL;DR: In this paper, a text on rings, fields and algebras is intended for graduate students in mathematics, aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation.

Abstract: This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

1,479 citations

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982 citations

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TL;DR: In this paper, it was shown that a commutative ring R is a finite ring if and only if the graph G associated with R is finitely colourable, and that the chromatic number of a graph G is the sum of the number of maximal ideals and number of units of R.

125 citations

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TL;DR: In this paper, the authors generalize the results to non-commutative rings and characterize all non-local ring R (not necessarily commutative) whose graph Γ(R) is a complete n-partite graph.

Abstract: Let R be a ring with unity. The graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Let Γ2(R) be the subgraph of Γ(R) induced by the non-unit elements of R. Let R be a commutative ring with unity and let J(R) denote the Jacobson radical of R. If R is not a local ring, then it was proved that:
(a)
If \(\Gamma_2(R)\backslash J(R)\) is a complete n-partite graph, then n = 2.
(b)
If there exists a vertex of \(\Gamma_2(R)\backslash J(R)\) which is adjacent to every vertex, then R ≅ ℤ2×F, where F is a field.
In this note we generalize the above results to non-commutative rings and characterize all non-local ring R (not necessarily commutative) whose \(\Gamma_2(R)\backslash J(R)\) is a complete n-partite graph.

19 citations

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TL;DR: In this article, the co-annihilating ideal graph of a commutative ring with identity was studied and its properties were investigated, where the vertex set is the set of all non-zero proper ideals of and two distinct vertices and are adjacent whenever.

Abstract: Let be a commutative ring with identity. The co-annihilating-ideal graph of , denoted by , is a graph whose vertex set is the set of all non-zero proper ideals of and two distinct vertices and are adjacent whenever . In this paper we initiate the study of the co-annihilating ideal graph of a commutative ring and we investigate its properties.

19 citations