# An infinite product of fields is not a semisimple ring

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).