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.
If $R$ is an integral domain, and every $R$-module is projective, must $R$ be a field?
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abstract-algebra
modules
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1Right, the product of fields is never an integral domain, so the only integral domains with your property are fields. – 2011-12-22
1 Answers
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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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1yes$I$can see that ... cause in general the ring would be semisimple artinian and then we need the domain property to conclude it is a field – 2017-07-06