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Let $D$ be an integral domain (zero ideal is prime). Then for every nonzero element $a,b \in D$, we have $\langle a\rangle\cap \langle b\rangle\neq 0$.

Now in a general case, let $R$ be a commutative ring with 1, such that $R$ has no nontrivial idempotent. I am looking for a general condition (other that $0$ is an irreducible ideal)  for $0$ (the zero ideal) under which $\langle a\rangle\cap \langle b\rangle\neq 0$ for all nonzero element $a,b \in R$.

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A partial answer:
It is easy to see that "for all nonzero elements $a$ and $b$, one has $\langle a\rangle\cap \langle b\rangle\neq 0$ iff for all nonzero ideals $I$ and $J$, has $I\cap J\neq 0.$" For Artinian local rings this is equivalent to saying that ring is Gorenstein; see Bruns_Herzog Exercise 3.2.15.

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