Example of $M,N$ modules that are not projectives but $M \otimes N$ is projective

Question

Show an example of $M,N$ modules that are not projectives but $M \otimes N$ is projective.

Consider $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/3\mathbb{Z}$ as $\mathbb{Z}$-modules, they are not projectives but $\mathbb{Z}/2\mathbb{Z} \otimes \mathbb{Z}/3\mathbb{Z} \simeq \{0 \}$ is projective.

Is it good?

From the [tag:solution-verification] tag wiki: "A question with this tag should include an explanation for why the argument presented is not convincing enough." — Shaun, Jan 14 '25 at 14:48

score 0 · Accepted Answer · answered Jan 14 '25 at 15:34

0

Your example is ok. Actually, you can always construct such examples with the facts that $\mathbb Z/n\mathbb Z$ is not a projective module for $n\geq 2$ and $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}\ $.

For your confusion in the comment, check the definition of the free module:

Let $M$ be an $R$-module. A subset $\mathcal{B}$ of $M$ is a basis if $\mathcal{B}$ is linearly independent and spans $M$. An $R$-module $M$ is said to be free if $M=\{0\}$ or if $M$ has a basis. Then we say that $M$ is free on $\mathcal{B}$.

answered Jan 14 '25 at 15:34

Bowei Tang

3,763

You don't need to make a case distinction, the zero module has a basis: the empty set. – Lukas Heger Jan 14 '25 at 16:46
@LukasHeger I'm following Hungerford's Algebra, that's how that book defines it, but your point of view is more appropriate. Thanks! – Bowei Tang Jan 15 '25 at 03:33

Example of $M,N$ modules that are not projectives but $M \otimes N$ is projective

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