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

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.