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!
2026-02-23 02:10:53.1771812653
An infinite product of fields is not a semisimple ring
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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).