Q:

Cartesian product of fields is semisimple iff the index set is finite

This question may have already been asked.

Let $R = \prod_{i\in I} K_i$ where each $K_i$ is a field. Show that
  $R$ is a semisimple ring iff the index set $I$ is finite.

I think I need to show that the direct product of a finite number of fields, when taken as a module over itself, can be reduced to corresponding to a direct sum of irreducible modules. Can I receive a hint in how to go about doing that?

A:

If you know that a semisimple ring is Artinian, then obviously $\prod_{i\in I} K_i$ is not Artinian if $I$ is infinite (just create an ascending chain of ideals using the factors you're given.)
If you know what an essential ideal is: $A=\oplus_{i\in I}K_i$ is an essential ideal in $\prod_{i\in I}K_i$, and if $I$ is infinite, $A$ is a proper ideal. But a proper essential ideal cannot be a direct summand. All ideals must be summands, of course, in a semisimple ring.

