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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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    Right, the product of fields is never an integral domain, so the only integral domains with your property are fields.2011-12-22

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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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    yes$I$can see that ... cause in general the ring would be semisimple artinian and then we need the domain property to conclude it is a field2017-07-06