The classification of rings with its genus of class of graphs

Authors : Thangaraj ASIR, Karuppiah MANO

Pages : 1424-1435

View : 15 | Download : 6

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let $R$ be a commutative ring, $I$ be a proper ideal of $R$, and $Sinsert ignore into journalissuearticles values(I);=\{a\in R : ra\in I \text{ for some } r\in R\sm I\}$ 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 $Tinsert ignore into journalissuearticles values(\Gamma_Iinsert ignore into journalissuearticles values(R););$, 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 $x+y\in Sinsert ignore into journalissuearticles values(I);$. In this paper, we determine all isomorphic classes of commutative Artinian rings whose ideal-based total graph has genus at most two.
Keywords : Commutative rings, total graph, planar, toroidal, genus