Geodesic
Some results on the annihilator graph of a commutative ring
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 151-169 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 ${\rm ann}_R(xy) \neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$, 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 $n \geq 1$.
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 ${\rm ann}_R(xy) \neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$, 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 $n \geq 1$.
DOI : 10.21136/CMJ.2017.0436-15
Classification : 05C75, 05C99, 13A99
Keywords: annihilator graph; zero-divisor graph; outerplanar; ring-graph; cut-vertex; clique number; weakly perfect; chromatic number; polynomial ring; ring of fractions 
@article{10_21136_CMJ_2017_0436_15,
     author = {Afkhami, Mojgan and Khashyarmanesh, Kazem and Rajabi, Zohreh},
     title = {Some results on the annihilator graph of a commutative ring},
     journal = {Czechoslovak Mathematical Journal},
     pages = {151--169},
     year = {2017},
     volume = {67},
     number = {1},
     doi = {10.21136/CMJ.2017.0436-15},
     mrnumber = {3633004},
     zbl = {06738510},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0436-15/}
}
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
SP  - 151
EP  - 169
VL  - 67
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0436-15/
DO  - 10.21136/CMJ.2017.0436-15
LA  - en
ID  - 10_21136_CMJ_2017_0436_15
ER  - 
%0 Journal Article
%A Afkhami, Mojgan
%A Khashyarmanesh, Kazem
%A Rajabi, Zohreh
%T Some results on the annihilator graph of a commutative ring
%J Czechoslovak Mathematical Journal
%D 2017
%P 151-169
%V 67
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0436-15/
%R 10.21136/CMJ.2017.0436-15
%G en
%F 10_21136_CMJ_2017_0436_15
Afkhami, Mojgan; Khashyarmanesh, Kazem; Rajabi, Zohreh. Some results on the annihilator graph of a commutative ring. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 151-169. doi: 10.21136/CMJ.2017.0436-15

[1] Afkhami, M.: When the comaximal and zero-divisor graphs are ring graphs and outerplanar. Rocky Mt. J. Math. 44 (2014), 1745-1761. | DOI | MR | JFM

[2] Afkhami, M., Barati, Z., Khashyarmanesh, K.: When the unit, unitary and total graphs are ring graphs and outerplanar. Rocky Mt. J. Math. 44 (2014), 705-716. | DOI | MR | JFM

[3] Akbari, S., Maimani, H. R., Yassemi, S.: When a zero-divisor graph is planar or a complete {$r$}-partite graph. J. Algebra 270 (2003), 169-180. | DOI | MR | JFM

[4] D. F. Anderson, M. C. Axtell, J. A. Stickles, Jr.: Zero-divisor graphs in commutative rings. Commutative Algebra. Noetherian and Non-Noetherian Perspectives M. Fontana et al. Springer, New York (2011), 23-45. | DOI | MR | JFM

[5] Anderson, D. F., Badawi, A.: On the zero-divisor graph of a ring. Commun. Algebra 36 (2008), 3073-3092. | DOI | MR | JFM

[6] Anderson, D. F., Badawi, A.: The total graph of a commutative ring. J. Algebra 320 (2008), 2706-2719. | DOI | MR | JFM

[7] Anderson, D. F., Levy, R., Shapiro, J.: Zero-divisor graphs, von Neumann regular rings, and Boolean algebras. J. Pure Appl. Algebra 180 (2003), 221-241. | DOI | MR | JFM

[8] Anderson, D. F., Livingston, P. S.: The zero-divisor graph of a commutative ring. J. Algebra 217 (1999), 434-447. | DOI | MR | JFM

[9] Anderson, D. D., Naseer, M.: Beck's coloring of a commutative ring. J. Algebra 159 (1993), 500-514. | DOI | MR | JFM

[10] Ashrafi, N., Maimani, H. R., Pournaki, M. R., Yassemi, S.: Unit graphs associated with rings. Commun. Algebra 38 (2010), 2851-2871. | DOI | MR | JFM

[11] Atiyah, M. F., Macdonald, I. G.: Introduction to Commutative Algebra. Series in Mathematics, Addison-Wesley Publishing Company, Reading, London (1969). | MR | JFM

[12] Badawi, A.: On the annihilator graph of a commutative ring. Commun. Algebra 42 (2014), 108-121. | DOI | MR | JFM

[13] Badawi, A.: On the dot product graph of a commutative ring. Commun. Algebra 43 (2015), 43-50. | DOI | MR | JFM

[14] Barati, Z., Khashyarmanesh, K., Mohammadi, F., Nafar, K.: On the associated graphs to a commutative ring. J. Algebra Appl. 11 (2012), 1250037, 17 pages. | DOI | MR | JFM

[15] Beck, I.: Coloring of commutative rings. J. Algebra 116 (1988), 208-226. | DOI | MR | JFM

[16] Belshoff, R., Chapman, J.: Planar zero-divisor graphs. J. Algebra 316 (2007), 471-480. | DOI | MR | JFM

[17] Coté, B., Ewing, C., Huhn, M., Plaut, C. M., Weber, D.: Cut-sets in zero-divisor graphs of finite commutative rings. Commun. Algebra 39 (2011), 2849-2861. | DOI | MR | JFM

[18] Gitler, I., Reyes, E., Villarreal, R. H.: Ring graphs and complete intersection toric ideals. Discrete Math. 310 (2010), 430-441. | DOI | MR | JFM

[19] Kelarev, A.: Graph Algebras and Automata. Pure and Applied Mathematics 257, Marcel Dekker, New York (2003). | MR | JFM

[20] Kelarev, A.: Labelled Cayley graphs and minimal automata. Australas. J. Comb. 30 (2004), 95-101. | MR | JFM

[21] Kelarev, A., Ryan, J., Yearwood, J.: Cayley graphs as classifiers for data mining: The influence of asymmetries. Discrete Math. 309 (2009), 5360-5369. | DOI | MR | JFM

[22] Maimani, H. R., Salimi, M., Sattari, A., Yassemi, S.: Comaximal graph of commutative rings. J. Algebra 319 (2008), 1801-1808. | DOI | MR | JFM

[23] West, D. B.: Introduction to Graph Theory. Prentice Hall, Upper Saddle River (1996). | MR | JFM

Cité par Sources :