How to show that the annihilating-ideal graph $AG(R)$ is an empty graph if and only if $R$ is an integral domain?

Question

While going through an article tittled, "On realizing zero-divisor graphs of po-semirings" by Houyi Yu and Tongsuo Wu, i learnt that the annihilating-ideal graph $AG(R)$ of a ring $R$, is a graph with vertex set $A(R)^*= \lbrace A(R)-\lbrace 0\rbrace\rbrace$, such that distinct vertices $I$ and $J$ are adjacent if and only if $IJ = 0$. Further, the articles says the graph $AG(R)$ is an empty graph if and only if $R$ is an integral domain. So, this is where my curiosity arises to know as an how the last statement holds? Also, i am not clear even the basic that how an integral domain or, any other rings or, semirings can be so closely associated with the graphs? Also, what could be the empty graph here? Does it mean a graph with vertices but no edges? or, a graph with no edges and no vertices? or, could be both?.

Answer 1

If $R$ is a nonzero commutative ring with $1$, then \begin{align*} R \text{ is an integral domain} & \iff \text{ the zero ideal is a prime ideal} \\ & \iff \text{ if the product of two ideals } IJ=0 \text{, then } I=0 \text{ or } J=0 \\ & \iff \text{ the zero ideal is the only annihilating-ideal} \\ & \iff \mathbb{AG}(R) \text{ has no vertices}. \end{align*}