# $A \oplus B$ a direct sum of $R$-modules. Possible to have a submodule $X$ of $A \oplus B$ s.t. $X \cap A = 0 = X \cap B$?

by Michael Vaughan   Last Updated November 18, 2018 15:20 PM

Let $$A \oplus B$$ be a direct sum of left $$R$$-modules. Is it possible to have a non-zero submodule $$X$$ of $$A \oplus B$$ such that $$X \cap A = 0 = X \cap B$$?

Perhaps not. A submodule of $$A \oplus B$$ would be of the form $$C \oplus D$$ with $$C \leq A$$, $$B \leq D$$, and both $$C$$ and $$D$$ would contain zero so wouldn't $$C \oplus \{0\} \subset C \oplus D \cap A$$?

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#### Answers 1

It depends on $$A$$ and $$B$$ (and $$R$$).

For example, if $$R$$ is a field and $$A,B$$ are nonzero $$R$$-vector spaces, then there are ($$1$$-dimensional) vector subspaces of $$A\oplus B$$ not contained in $$A$$ or $$B$$.

On the other hand, if $$R=\mathbb{Z}$$ and $$A=\mathbb{Z}/2\mathbb{Z}$$, $$B=\mathbb{Z}/3\mathbb{Z}$$, then the only nontrivial submodules of $$A\oplus B=\mathbb{Z}/6\mathbb{Z}$$ are $$A,B$$ and $$A\oplus B$$.