Let $A$ be a noetherian ring such that $2$ is not a zero divisor of $A$. Let $M$ be a submodule of the free module $F=A^r$. Clearly $2$ is not a zero divisor of $M$, i.e. $2x=0$ implies $x=0$ for all $x \in M$. But how is the situation for the exterior power $\bigwedge^2 M$? Can $2$ be a zero divisor nevertheless? If yes, what are suitable conditions on $A$ or $M$ such that this can not happen? (Of course if $2$ is invertible in $A$, but I don't want to assume that.)
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