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

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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.

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