On Graphs Related to Comaximal Ideals of a Commutative Ring
Keyword(s):
The Core
◽
We study the co maximal graph Ω(R), the induced subgraph Γ(R) of Ω(R) whose vertex set is R∖(U(R)∪J(R)), and a retract Γr(R) of Γ(R), where R is a commutative ring. For a graph Γ(R) which contains a cycle, we show that the core of Γ(R) is a union of triangles and rectangles, while a vertex in Γ(R) is either an end vertex or a vertex in the core. For a nonlocal ring R, we prove that both the chromatic number and clique number of Γ(R) are identical with the number of maximal ideals of R. A graph Γr(R) is also introduced on the vertex set {Rx∣x∈R∖(U(R)∪J(R))}, and graph properties of Γr(R) are studied.
Keyword(s):
Keyword(s):
2015 ◽
Vol 14
(06)
◽
pp. 1550079
◽
Keyword(s):
2012 ◽
Vol 11
(06)
◽
pp. 1250114
◽
Keyword(s):
2016 ◽
Vol 16
(07)
◽
pp. 1750132
◽
Keyword(s):
2016 ◽
Vol 15
(07)
◽
pp. 1650124
◽
Keyword(s):
2020 ◽
Vol 12
(03)
◽
pp. 2050023
Keyword(s):
2018 ◽
Vol 10
(04)
◽
pp. 1850047
Keyword(s):
2013 ◽
Vol 12
(04)
◽
pp. 1250199
◽