On the planarity and perfectness of annihilator ideal graphs

  • Reza Nikandish
  • M. J Nikmehr
  • S. M Hosseini

Abstract

Let $R$ be a commutative ring with unity. The annihilator ideal graph of $R$, denoted by $\Gamma _{\mathrm{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 \mathrm{Ann} _{R} (J) \neq \lbrace 0\rbrace $ or $J \cap \mathrm{Ann} _{R} (I) \neq \lbrace 0\rbrace $.
In this paper, all rings with planar annihilator ideal graphs are classified.
Furthermore, we show that all annihilator ideal graphs are perfect. Among other results, it is proved that if $\Gamma _{\mathrm{Ann}} (R) $ is a tree, then $\Gamma _{\mathrm{Ann}} (R) $ is star.

Published
2019-12-30
How to Cite
Nikandish, R., Nikmehr, M. J., & Hosseini, S. M. (2019). On the planarity and perfectness of annihilator ideal graphs. Tamkang Journal of Mathematics, 50(4), 361-369. https://doi.org/10.5556/j.tkjm.50.2019.2707
Section
Papers