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.

  • $\begingroup$ Can you say more about that equivalence? It's not obvious to me. Or is this a standard fact? $\endgroup$ – Charles Hudgins Jan 31 at 22:20
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    $\begingroup$ @CharlesHudgins Every non-zero ideal contains a non-zero principal ideal. Therefore that intersection is equal to the intersection of all the non-zero principal ideals. $\endgroup$ – Gae. S. Jan 31 at 22:22

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.


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