THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING Article Swipe

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Abolfazl Alibemani , M. Bakhtyiari , R. Nikandish , M. J. Nikmehr ·

YOU? · · 2015 · Open Access · · DOI: https://doi.org/10.4134/jkms.2015.52.2.417

Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by ${\Gamma}_{Ann}(R)$, is a graph whose vertices are all non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if $I{\cap}Ann(J){\neq}\{0\}$ or $J{\cap}Ann(I){\neq}\{0\}$. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose annihilator ideal graphs are totally disconnected. Also, we study diameter, girth, clique number and chromatic number of this graph. Moreover, we study some relations between annihilator ideal graph and zero-divisor graph associated with R. Among other results, it is proved that for a Noetherian ring R if ${\Gamma}_{Ann}(R)$ is triangle free, then R is Gorenstein.

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Mathematics Annihilator Commutative ring Combinatorics Primary ideal Discrete mathematics Clique number Graph Noetherian ring Associated prime Zero divisor Principal ideal ring Commutative property Pure mathematics Algebra over a field Prime (order theory)

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Type: article
Language: en
Landing Page: https://doi.org/10.4134/jkms.2015.52.2.417
PDF: http://koreascience.or.kr:80/article/JAKO201508556261674.pdf
OA Status: diamond
Cited By: 12
References: 6
Related Works: 10
OpenAlex ID: https://openalex.org/W2320813351

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DOI: https://doi.org/10.4134/jkms.2015.52.2.417

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Title: THE ANNIHILATOR IDEAL GRAPH OF A COMMUTATIVE RING

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Language: en

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Publication year: 2015

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Publication date: 2015-03-01

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Authors: Abolfazl Alibemani, M. Bakhtyiari, R. Nikandish, M. J. Nikmehr

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Landing page: https://doi.org/10.4134/jkms.2015.52.2.417

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Concepts: Mathematics, Annihilator, Commutative ring, Combinatorics, Primary ideal, Discrete mathematics, Clique number, Graph, Noetherian ring, Associated prime, Zero divisor, Principal ideal ring, Commutative property, Pure mathematics, Algebra over a field, Prime (order theory)

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Cited by: 12

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