Year 2018, Volume 5 , Issue 2, Pages 85 - 99 2018-05-15

Some results on the comaximal ideal graph of a commutative ring

Subramanian VİSWESWARAN [1] , Jaydeep PAREJİYA [2]


The rings considered in this article are commutative with identity which admit at least two maximal ideals. Let $R$ be a ring such that $R$ admits at least two maximal ideals. Recall from Ye and Wu (J. Algebra Appl. 11(6): 1250114, 2012) that the comaximal ideal graph of $R$, denoted by $\mathscr{C}(R)$ is an undirected simple graph whose vertex set is the set of all proper ideals $I$ of $R$ such that $I\not\subseteq J(R)$, where $J(R)$ is the Jacobson radical of $R$ and distinct vertices $I_{1}$, $I_{2}$ are joined by an edge in $\mathscr{C}(R)$ if and only if $I_{1} + I_{2} = R$. In Section 2 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is planar. In Section 3 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is a split graph. In Section 4 of this article, we classify rings $R$ such that $\mathscr{C}(R)$ is complemented and moreover, we determine the $S$-vertices of $\mathscr{C}(R)$.
Comaximal ideal graph, Special principal ideal ring, Planar graph, Split graph, Complement of a vertex in a graph
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Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Orcid: 0000-0002-4905-809X
Author: Subramanian VİSWESWARAN (Primary Author)

Orcid: 0000-0002-2072-2719
Author: Jaydeep PAREJİYA

Dates

Publication Date : May 15, 2018

Bibtex @research article { jacodesmath423751, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2018}, volume = {5}, pages = {85 - 99}, doi = {10.13069/jacodesmath.423751}, title = {Some results on the comaximal ideal graph of a commutative ring}, key = {cite}, author = {Vi̇sweswaran, Subramanian and Pareji̇ya, Jaydeep} }
APA Vi̇sweswaran, S , Pareji̇ya, J . (2018). Some results on the comaximal ideal graph of a commutative ring . Journal of Algebra Combinatorics Discrete Structures and Applications , 5 (2) , 85-99 . DOI: 10.13069/jacodesmath.423751
MLA Vi̇sweswaran, S , Pareji̇ya, J . "Some results on the comaximal ideal graph of a commutative ring" . Journal of Algebra Combinatorics Discrete Structures and Applications 5 (2018 ): 85-99 <https://dergipark.org.tr/en/pub/jacodesmath/issue/37143/423751>
Chicago Vi̇sweswaran, S , Pareji̇ya, J . "Some results on the comaximal ideal graph of a commutative ring". Journal of Algebra Combinatorics Discrete Structures and Applications 5 (2018 ): 85-99
RIS TY - JOUR T1 - Some results on the comaximal ideal graph of a commutative ring AU - Subramanian Vi̇sweswaran , Jaydeep Pareji̇ya Y1 - 2018 PY - 2018 N1 - doi: 10.13069/jacodesmath.423751 DO - 10.13069/jacodesmath.423751 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 85 EP - 99 VL - 5 IS - 2 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.423751 UR - https://doi.org/10.13069/jacodesmath.423751 Y2 - 2018 ER -
EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications Some results on the comaximal ideal graph of a commutative ring %A Subramanian Vi̇sweswaran , Jaydeep Pareji̇ya %T Some results on the comaximal ideal graph of a commutative ring %D 2018 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 5 %N 2 %R doi: 10.13069/jacodesmath.423751 %U 10.13069/jacodesmath.423751
ISNAD Vi̇sweswaran, Subramanian , Pareji̇ya, Jaydeep . "Some results on the comaximal ideal graph of a commutative ring". Journal of Algebra Combinatorics Discrete Structures and Applications 5 / 2 (May 2018): 85-99 . https://doi.org/10.13069/jacodesmath.423751
AMA Vi̇sweswaran S , Pareji̇ya J . Some results on the comaximal ideal graph of a commutative ring. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018; 5(2): 85-99.
Vancouver Vi̇sweswaran S , Pareji̇ya J . Some results on the comaximal ideal graph of a commutative ring. Journal of Algebra Combinatorics Discrete Structures and Applications. 2018; 5(2): 85-99.

Authors of the Article
Subramanian VİSWESWARAN [1]
Jaydeep PAREJİYA [2]