Journal of the
Korean Mathematical Society JKMS

Let $R$ be a commutative ring and $I$ its proper ideal, let $S(I)$ be the set of all elements of $R$ that are not prime to $I$. Here we introduce and study the total graph of a commutative ring $R$ with respect to proper ideal $I$, denoted by $T(\Gamma_{I}(R))$. It is the (undirected) 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 $x+y\in S(I)$. The total graph of a commutative ring, that denoted by $T(\Gamma(R))$, is the graph where the vertices are all elements of $R$ and where there is an undirected edge between two distinct vertices $x$ and $y$ if and only if $x+y\in Z(R)$ which is due to Anderson and Badawi [2]. In the case $I=\{0\}$, $T(\Gamma_{I}(R))= T(\Gamma(R))$; this is an important result on the definition.

Keywords: commutative rings, zero divisor, total graph

MSC numbers: 13D45, 13E10, 13C05