coprime ideals in a ring

Question

Suppose $R$ is a ring ($R$ may not have a unit and can be non-commutative), $I,J$ are two nonzero proper ideals in $R$ such that $I+J=R$ and $I\cap J\neq 0$. I wonder if there exists a possibility that $I$ is essential in $R$? ($\{r\in R:rI=Ir=0\}=\{0\}$)

Answer 1

Maybe I misunderstand but try $2\mathbb{Z}$ and $3\mathbb{Z}$ in $\mathbb{Z}$. They are coprime and essential in the sense you defined.