Turkish Journal of Mathematics
Abstract
Let $R$ be a commutative ring, $I$ be a proper ideal of $R$, and $S(I)=\{a\in R : ra\in I \text{ for some } r\in R\sm I\}$ be the set of all elements of $R$ that are not prime to $I$. The total graph of $R$ with respect to $I$, denoted by $T(\Gamma_I(R))$, is the simple graph with all elements of $R$ as vertices, and for distinct $x,y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x+y\in S(I)$. In this paper, we determine all isomorphic classes of commutative Artinian rings whose ideal-based total graph has genus at most two.
DOI
10.3906/mat-1710-34
Keywords
Commutative rings, total graph, planar, toroidal, genus
First Page
1424
Last Page
1435
Recommended Citation
ASIR, THANGARAJ and MANO, KARUPPIAH
(2018)
"The classification of rings with its genus of class of graphs,"
Turkish Journal of Mathematics: Vol. 42:
No.
3, Article 53.
https://doi.org/10.3906/mat-1710-34
Available at:
https://journals.tubitak.gov.tr/math/vol42/iss3/53
