# When is intersection of infinitely many maximal ideals zero?

I've been trying without success to figure out what are the rings $R$ such that whenever $M_n, n \in \omega$ is a countably infinite collection of pairwise distinct maximal ideals then $\bigcap_{n \in \omega}M_n=0$. If $R$ is a Dedekind domain then this obviously holds, and if $R$ has this property and has infinitely many maximal ideals then it has to have zero radical. Thanks for any input or hint.

-
What if all ideals are equal ? –  Amr Dec 2 '12 at 11:11
Maximal ideals $M_n$ are supposed to be distinct, I should have probably written pairwise distinct. –  Fred.Fred Dec 2 '12 at 11:15
A related notion is a semiprimitive ring, which is a ring such that the Jacobson radical is zero. –  JSchlather Dec 2 '12 at 11:18
The set $\{0,1,2,\cdots\}$. –  Fred.Fred Dec 2 '12 at 12:05

The condition as you've written it basically just characterizes rings which admit a faithful semisimple module of "countable length," that is, it is of the form $\oplus_{i\in\omega}S_i$ with the $S_i$ all simple.
Since it has Jacobson radical zero, it is a subclass of the semiprimitive rings (they admit faithful semisimple modules, but depending on the ring they have to have more than $\omega$ summands.)
Here are some examples: On one hand, the ring could be very "wide". Take for example, $\prod_{i\in \kappa} \Bbb F$ where $\kappa$ is your favorite infinite cardinal. With $\kappa=\omega$, you have a ring with the condition you are interested in. With $\kappa>\omega$, you will not be able to find countably many maximal ideals which intersect to zero.
Another example is the endomorphism ring of $\oplus_{i\in \kappa}\Bbb F$ for some field $\Bbb F$. When $\kappa=\omega$ you get "sqaure" matrices with rows and columns indexed by $\omega$, each of whose columns have finitely many nonzero entries. You can check that if you form the collection $I_i$ of matrices whose $i$th row are all zero, $I_i$ is a maximal left ideal of the ring, and $\cap_{i\in \omega} I_i=\{0\}$. If $\kappa>\omega$, then it will be impossible for countably many maximal right ideals to intersect to zero (since then it would have to be isomorphic to the ring with $\kappa=\omega$, but that's impossible due to cardinality reasons.)