TY - JOUR
AU - Nikandish, Reza
AU - Nikmehr, M. J
AU - Hosseini, S. M
PY - 2019/12/30
Y2 - 2020/09/23
TI - On the planarity and perfectness of annihilator ideal graphs
JF - Tamkang Journal of Mathematics
JA - Tamkang J. Math.
VL - 50
IS - 4
SE -
DO - 10.5556/j.tkjm.50.2019.2707
UR - https://journals.math.tku.edu.tw/index.php/TKJM/article/view/2707
SP - 361-369
AB - <p>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<br />$ I \cap \mathrm{Ann} _{R} (J)
eq \lbrace 0\rbrace $ or $J \cap \mathrm{Ann} _{R} (I)
eq \lbrace 0\rbrace $.<br />In this paper, all rings with planar annihilator ideal graphs are classified.<br />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.</p>
ER -