Question

Let {F_i : i ∈ I} be a family of fields. Show that the direct product R =(direct product)F_i is a semisimple ring iff the indexing set I is finite.

Answer 1

First suppose $R$ is semisimple. It is obvious that $I$ must be finite. Conversely, assume $I$ is finite, and consider any ideal $A \subseteq R$. Then, $A = \oplus_{i \in J} F_i$ for a subset $J \subseteq I$. Clearly, $A$ is a direct summand of $R$ and, so $R$ is a semisimple module, as desired.