I want to show that an infinite product of fields is not a semisimple ring. I know Artin-Wedderburn Theorem, but I wonder can I explain it without using this theorem? Any help will be appreciated. Thanks!
Any ideal $I$ in a semisimple ring $R$ is principal, since the inclusion map $I\to R$ is split by a surjective module-homomorphism $R\to I$. But in any infinite product of nonzero rings, the ideal of elements of finite support is not principal (or even finitely generated).
If you know semisimple rings are Artinian and Noetherian, another way would be to show an infinite product of nonzero rings is not Artinian or not Noetherian using chains of ideals. Either one is equally easy to show.