In Keith Conrad's notes: https://kconrad.math.uconn.edu/blurbs/linmultialg/extmod.pdf
Theorem 2.10 reads:
Let $k\geq 2$. If $2\in R^\times$, then a multilinear function $f:M^k \to N$ which is skew-symmetric is alternating.
After the theorem he says
Strictly speaking, the assumption that $2\in R^\times$ could be weakened to $2$ not being a zero divisor in $R$.
Why is this condition not "$2\notin\text{ann}_R(x)$ for any $x\in N$", i.e., $2$ is not a zerodivisor of $N$? I thought the key line is $$2f(m,m,m_3,\ldots,m_k)=0\implies f(m,m,m_3,\ldots,m_k)=0$$ (which is true if $2\in R^\times$).
However, if $R=\mathbb{Z}$ and $N=\mathbb{Z}/4\mathbb{Z}$, then if $f(m,m,m_3,\ldots,m_k)=2$, $2f(m,m,m_3,\ldots,m_k)=0\;\not\!\!\!\implies f(m,m,m_3,\ldots,m_k)=0$, even though $2$ is a nonzerodivisor of $R$ (but here $2$ is a zerodivisor of $N$).
But Wikipedia backs up his point, and it's Keith Conrad, so I'm probably missing something obvious or confusing definitions...