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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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Yes.… – Qiaochu Yuan Dec 22 '11 at 19:18
A key search term is 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
up vote 8 down vote accepted

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.

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@jspecter: Corrected. Thanks! – Pierre-Yves Gaillard Dec 22 '11 at 19:25

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