Let $M$ be a module over a commutative ring $R$, and let $v_0,\dots,v_{k-1}$ be elements of $M$. If $R$ is a field then $v_0\wedge\dots\wedge v_{k-1}$ is equal to $0$ if and only if $v_0,\dots,v_{k-1}$ are linearly dependent. But if $R$ isn't a field then this needn't be true.
For example if we view $\frac{\mathbb{Z}}{2\mathbb{Z}}$ as a $\mathbb Z$-module then $1\in\frac{\mathbb{Z}}{2\mathbb{Z}}$ is linearly dependent on its own, because $2.1=0$, but it doesn't get sent to $0$ in $\Lambda^1\left(\frac{\mathbb{Z}}{2\mathbb{Z}}\right)$.
Is there a nice characterisation of which lists of vectors do get sent to $0$?