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Let M and N be left R-Modules, is it possible to construct an example of a sub-module of $M \oplus N$ that is not a direct sum of a submodule of M and a submodule of N?

I don't know a whole lot of Modules so I was trying to think of ideals. Maybe some polynomials in two variables x,y?

Anyone think this is on track?

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    Try the direct sum of $\mathbb{R}$-modules $\mathbb{R} \oplus \mathbb{R}$.2011-09-11

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Hint: Let $R=N=M=\mathbb{Z}$. There are many $(x,y)\in \mathbb{Z}\oplus \mathbb{Z}$ such that the submodule generated by $(x,y)$ is not the direct sum of submodules of $\mathbb{Z}$. Can you find some?

This works in much greater generality than $\mathbb{Z}$, of course.

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    Actually, any submodule of a direct sum $M\oplus N$ is contained in the direct sum of two submodules: namely, $M\oplus N$ (because $M$ is a submodule of itself, and $N$ is a submodule of itself, $M\oplus N$ is a direct sum of submodules).2011-09-12