Some results on the annihilator graph of a commutative ring

Mojgan Afkhami; Kazem Khashyarmanesh; Zohreh Rajabi

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 151-169
  • ISSN: 0011-4642

Abstract

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Let R be a commutative ring. The annihilator graph of R , denoted by AG ( R ) , is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann R ( x y ) ≠ ann R ( x ) ∪ ann R ( y ) , where for z ∈ R , ann R ( z ) = { r ∈ R : r z = 0 } . In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1 , 2 or 3 . Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG ( R ) and AG ( T ( R ) ) are isomorphic, where T ( R ) is the total quotient ring of R . Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n , where n ≥ 1 .

How to cite

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Afkhami, Mojgan, Khashyarmanesh, Kazem, and Rajabi, Zohreh. "Some results on the annihilator graph of a commutative ring." Czechoslovak Mathematical Journal 67.1 (2017): 151-169. <http://eudml.org/doc/287889>.

@article{Afkhami2017,
abstract = {Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by $\{\rm AG\}(R)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if $\{\rm ann\}_R(xy) \ne \{\rm ann\}_R(x)\cup \{\rm ann\}_R(y)$, where for $z \in R$, $\{\rm ann\}_R(z) = \lbrace r \in R \colon rz = 0\rbrace $. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator graph under the extension of $R$ to polynomial rings and rings of fractions. For instance, we show that the graphs $\{\rm AG\}(R)$ and $\{\rm AG\}(T(R))$ are isomorphic, where $T(R)$ is the total quotient ring of $R$. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo $n$, where $n \ge 1$.},
author = {Afkhami, Mojgan, Khashyarmanesh, Kazem, Rajabi, Zohreh},
journal = {Czechoslovak Mathematical Journal},
keywords = {annihilator graph; zero-divisor graph; outerplanar; ring-graph; cut-vertex; clique number; weakly perfect; chromatic number; polynomial ring; ring of fractions},
language = {eng},
number = {1},
pages = {151-169},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Some results on the annihilator graph of a commutative ring},
url = {http://eudml.org/doc/287889},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Afkhami, Mojgan
AU - Khashyarmanesh, Kazem
AU - Rajabi, Zohreh
TI - Some results on the annihilator graph of a commutative ring
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 151
EP - 169
AB - Let $R$ be a commutative ring. The annihilator graph of $R$, denoted by ${\rm AG}(R)$, is the undirected graph with all nonzero zero-divisors of $R$ as vertex set, and two distinct vertices $x$ and $y$ are adjacent if and only if ${\rm ann}_R(xy) \ne {\rm ann}_R(x)\cup {\rm ann}_R(y)$, where for $z \in R$, ${\rm ann}_R(z) = \lbrace r \in R \colon rz = 0\rbrace $. In this paper, we characterize all finite commutative rings $R$ with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings $R$ whose annihilator graphs have clique number $1$, $2$ or $3$. Also, we investigate some properties of the annihilator graph under the extension of $R$ to polynomial rings and rings of fractions. For instance, we show that the graphs ${\rm AG}(R)$ and ${\rm AG}(T(R))$ are isomorphic, where $T(R)$ is the total quotient ring of $R$. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo $n$, where $n \ge 1$.
LA - eng
KW - annihilator graph; zero-divisor graph; outerplanar; ring-graph; cut-vertex; clique number; weakly perfect; chromatic number; polynomial ring; ring of fractions
UR - http://eudml.org/doc/287889
ER -

References

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