# If $R$ is an integral domain, and every $R$-module is projective, must $R$ be a field?

Let $R$ be an integral domain with the property that all modules over $R$ are projective. Does it follow that $R$ is a field? Obviously the converse is true.

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– Qiaochu Yuan Dec 22 '11 at 19:18
A key search term is Global Dimension. en.wikipedia.org/wiki/Global_dimension – jspecter Dec 22 '11 at 19:22
But the product of fields is not necessarily a field, is it? Does specifying that $R$ be integral make it the trivial product? – Lepidopterist Dec 22 '11 at 19:22
If this is true I would like to know a little bit about how technical the proof is. I have tried proving this using basic facts about projective modules but can't seem to do it. – Lepidopterist Dec 22 '11 at 19:24
Right, the product of fields is never an integral domain, so the only integral domains with your property are fields. – Thomas Andrews Dec 22 '11 at 19:32

If $R$ is not a field, it has a nonzero proper ideal $I$, and $R/I$ is not projective, because it is a nonzero torsion module.
Variation: The canonical projection $R\to R/I$ doesn't split.