# Rings with non-zero intersection of all non-zero ideals

Let $$R$$ be a ring such that $$\bigcap_{I\neq 0} I \neq 0$$, ie. has a non-zero intersection of all non-zero ideals. This is equivalent to ask for the existence of an element $$a\in R$$ which is a multiple of all non-zero elements of the ring.

What can be said of such rings? Can they be classified? Right now the only examples I can think of are $$\Bbb Z/(p^k)$$ and fields. I also know that if $$R$$ is an integral domain with this property then it must be a field. Are there other examples?