TY - JOUR
ID - 13644
TI - Some results on the complement of a new graph associated to a commutative ring
JO - Communications in Combinatorics and Optimization
JA - CCO
LA - en
SN - 2538-2128
AU - Visweswaran, S.
AU - Parmar, Anirudhdha
AD - Saurashtra University
Y1 - 2017
PY - 2017
VL - 2
IS - 2
SP - 119
EP - 138
KW - Annihilating ideal of a ring
KW - maximal N-prime of (0)
KW - connected graph
KW - diameter. girth
DO - 10.22049/cco.2017.25908.1053
N2 - The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let $R$ be a ring. We denote the collection of all ideals of $R$ by $mathbb{I}(R)$ and $mathbb{I}(R)backslash {(0)}$ by $mathbb{I}(R)^{*}$. Alilou et al. [A. Alilou, J. Amjadi and S.M. Sheikholeslami, {em A new graph associated to a commutative ring}, Discrete Math. Algorithm. Appl. {bf 8} (2016) Article ID: 1650029 (13 pages)] introduced and investigated a new graph associated to $R$, denoted by $Omega_{R}^{*}$ which is an undirected graph whose vertex set is $mathbb{I}(R)^{*}backslash {R}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if either $(Ann_{R}I)J = (0)$ or $(Ann_{R}J)I = (0)$. Several interesting theorems were proved on $Omega_{R}^{*}$ in the aforementioned paper and they illustrate the interplay between the graph-theoretic properties of $Omega_{R}^{*}$ and the ring-theoretic properties of $R$. The aim of this article is to investigate some properties of $(Omega_{R}^{*})^{c}$, the complement of the new graph $Omega_{R}^{*}$ associated to $R$.
UR - http://comb-opt.azaruniv.ac.ir/article_13644.html
L1 - http://comb-opt.azaruniv.ac.ir/article_13644_1b27eaa14546119e0ee5915425b1cb0b.pdf
ER -