$A \oplus B$ 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 $C \oplus \{0\} \subset C \oplus D \cap A$?

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 $A\oplus B$.

November 18, 2018 14:59 PM

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