# coprime ideals in a ring

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 $$R$$? ($$\{r\in R:rI=Ir=0\}=\{0\}$$)

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.