R is a ring with 1, and M, N two modules over R .
Suppose that M is isomorphic to some submodule of N, and N is isomorphic to some submodule of M, then is M isomorphic to N?
2025-01-13 08:02:14.1736755334
A problem about isomorphism and submodules
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No. Consider any integral domain with a non-principal ideal. For argument, say $R=\Bbb C[X,Y]$ and $I=\left<X,Y\right>$. Then as $R$-modules, $R$ contains $I$ which contains $\left<a\right>$ for any nonzero element of $R$. As $\left<a\right>$ is isomorphic as a module to $R$, then we have two non-isomorphic modules $R$ and $I$ each isomorphic to a submodule of the other.