Let $R$ be an integral domain and $M$ be a free module over $R$.
My question is this:
Is the intersection of two free submodules of $M$ a free submodule?
P.S: We need the intersection of two free submodules is not $\{0\}$.
2026-03-25 21:22:12.1774473732
In a free module, must the intersection of two free submodules be a free submodule?
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Let $P$ be a projective module which is not free over a domain $R$. (There are such domains.) We know that we may complement $P$ with another module so that $P\oplus N$ is a free module.
Consider the module $M=N\oplus P\oplus N\oplus P$ which is an (external) direct sum of two free $R$ modules, so it is free. Clearly $N\oplus P\oplus\{0\}\oplus\{0\}\cong \{0\}\oplus P\oplus N\oplus\{0\}$ are free modules. But their intersection is, you guessed it, $\{0\}\oplus P\oplus\{0\}\oplus\{0\}\cong P$, not free.