Open AccessOn the planarity and perfectness of annihilator ideal graphs
Author(s) -
Reza Nikandish,
M. J. Nikmehr,
Seyed Mehrdad Hosseini
Publication year - 2019
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.50.2019.2707
Subject(s) - annihilator , mathematics , planarity testing , combinatorics , ideal (ethics) , commutative ring , graph , discrete mathematics , commutative property , pure mathematics , algebra over a field , philosophy , epistemology
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.