Some classes of perfect strongly annihilating-ideal graphs associated with commutative rings

Mitra Jalali; Abolfazl Tehranian; Reza Nikandish; Hamid Rasouli

Commentationes Mathematicae Universitatis Carolinae (2020)

  • Volume: 61, Issue: 1, page 27-34
  • ISSN: 0010-2628

Abstract

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Let R be a commutative ring with identity and A ( R ) be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of R is defined as the graph SAG ( R ) with the vertex set A ( R ) * = A ( R ) ∖ { 0 } and two distinct vertices I and J are adjacent if and only if I ∩ Ann ( J ) ≠ ( 0 ) and J ∩ Ann ( I ) ≠ ( 0 ) . In this paper, the perfectness of SAG ( R ) for some classes of rings R is investigated.

How to cite

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Jalali, Mitra, et al. "Some classes of perfect strongly annihilating-ideal graphs associated with commutative rings." Commentationes Mathematicae Universitatis Carolinae 61.1 (2020): 27-34. <http://eudml.org/doc/297380>.

@article{Jalali2020,
abstract = {Let $R$ be a commutative ring with identity and $A(R)$ be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of $R$ is defined as the graph $\{\rm SAG\}(R)$ with the vertex set $A(R)^*=A(R)\setminus \lbrace 0\rbrace $ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap \{\rm Ann\}(J)\ne (0)$ and $J\cap \{\rm Ann\}(I)\ne (0)$. In this paper, the perfectness of $\{\rm SAG\}(R)$ for some classes of rings $R$ is investigated.},
author = {Jalali, Mitra, Tehranian, Abolfazl, Nikandish, Reza, Rasouli, Hamid},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {strongly annihilating-ideal graph; perfect graph; chromatic number; clique number},
language = {eng},
number = {1},
pages = {27-34},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Some classes of perfect strongly annihilating-ideal graphs associated with commutative rings},
url = {http://eudml.org/doc/297380},
volume = {61},
year = {2020},
}

TY - JOUR
AU - Jalali, Mitra
AU - Tehranian, Abolfazl
AU - Nikandish, Reza
AU - Rasouli, Hamid
TI - Some classes of perfect strongly annihilating-ideal graphs associated with commutative rings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2020
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 61
IS - 1
SP - 27
EP - 34
AB - Let $R$ be a commutative ring with identity and $A(R)$ be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of $R$ is defined as the graph ${\rm SAG}(R)$ with the vertex set $A(R)^*=A(R)\setminus \lbrace 0\rbrace $ and two distinct vertices $I$ and $J$ are adjacent if and only if $I\cap {\rm Ann}(J)\ne (0)$ and $J\cap {\rm Ann}(I)\ne (0)$. In this paper, the perfectness of ${\rm SAG}(R)$ for some classes of rings $R$ is investigated.
LA - eng
KW - strongly annihilating-ideal graph; perfect graph; chromatic number; clique number
UR - http://eudml.org/doc/297380
ER -

References

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  1. Aalipour G., Akbari S., Nikandish R., Nikmehr M. J., Shaveisi F., 10.1016/j.disc.2011.10.020, Discrete Math. 312 (2012), no. 17, 2620–2626. MR2935413DOI10.1016/j.disc.2011.10.020
  2. Anderson D. F., Livingston P. S., 10.1006/jabr.1998.7840, J. Algebra 217 (1999), no. 2, 434–447. Zbl1035.13004MR1700509DOI10.1006/jabr.1998.7840
  3. Badawi A., 10.1080/00927872.2012.707262, Comm. Algebra 42 (2014), no. 1, 108–121. MR3169557DOI10.1080/00927872.2012.707262
  4. Beck I., 10.1016/0021-8693(88)90202-5, J. Algebra 116 (1988), no. 1, 208–226. Zbl0654.13001MR0944156DOI10.1016/0021-8693(88)90202-5
  5. Behboodi M., Rakeei Z., 10.1142/S0219498811004896, J. Algebra Appl. 10 (2011), no. 4, 727–739. MR2834112DOI10.1142/S0219498811004896
  6. Chudnovsky M., Robertson N., Seymour P., Thomas R., 10.4007/annals.2006.164.51, Ann. of Math. (2) 164 (2006), no. 1, 51–229. MR2233847DOI10.4007/annals.2006.164.51
  7. Diestel R., Graph Theory, Graduate Texts in Mathematics, 173, Springer, New York, 2000. Zbl1218.05001MR1743598
  8. Nikandish R., Nikmehr M. J., Tohidi N. Kh., 10.1007/s40590-017-0179-1, Bol. Soc. Mat. Mex. (3) 24 (2018), no. 2, 307–318. MR3854466DOI10.1007/s40590-017-0179-1
  9. Sharp R. Y., Steps in Commutative Algebra, London Mathematical Society Student Texts, 51, Cambridge University Press, Cambridge, 2000. MR1817605
  10. Taheri R., Behboodi M., Tehranian A., The spectrum subgraph of the annihilating-ideal graph of a commutative ring, J. Algebra Appl. 14 (2015), no. 8, 1550130, 19 pages. MR3339815
  11. Tohidi N. Kh., Nikmehr M. J., Nikandish R., 10.1142/S1793830917500288, Discrete Math. Algorithms Appl. 9 (2017), no. 2, 1750028, 13 pages. MR3635064DOI10.1142/S1793830917500288
  12. West D. B., Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996. Zbl1121.05304MR1367739
  13. Wisbauer R., Foundations of Module and Ring Theory, Algebra, Logic and Applications, 3, Gordon and Breach Science Publishers, Philadelphia, 1991. Zbl0746.16001MR1144522

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