TITLE: Rings with non-zero intersection of all non-zero ideals
QUESTION [7 upvotes]: 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?
Thank you in advance.

REPLY [3 votes]: This is equivalent to being a subdirectly irreducible ring. (See Lam’s First course in noncommutative rings p 192, for example.
Such rings are not expressable as a subdirect product of two nonzero rings.