I know that the condition is an "if and only if". I've proved that if every $R$ module is free then $R$ is a field, but I can't see so clearly the converse.
2026-03-25 03:22:46.1774408966
Why does every module over an $R$ conmutative ring is free, if $R$ is a field?
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Remember that a module over a field is usually better known as a vector space. While you might not have had this terminology back when you were studying them, 'Every module over a field is free," just means, "Every vector space has a basis," and you would definitely have proven that for finite dimensional ones back then.
Though... saying that every vector space has a basis means getting into Axiom of Choice territory (they're equivalent, I think) so this isn't actually true if you don't accept the axiom of choice.