Does $M \otimes N $ finitely generated imply that $M$ and $N$ are also finitely generated? I know that the converse is true, but I'm not really sure about this one. Thanks.
2026-03-30 16:41:39.1774888899
Finitely generated Tensor Product
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No : there are examples where neither is finitely generated, and examples where one of them is and not the other: take for instance $\mathbb{Q}\otimes\text{(any torsion abelian group)}$, which is $0$.
Now it's known that there are finitely generated and infinitely generated torsion abelian groups, so that's it.