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...

  • $\begingroup$ Sorry, I have had tensors in mind lately and that influenced my answer. I'll think about it more. Maybe there is an inaccuracy or missing assumption after all. $\endgroup$ – rschwieb Feb 22 at 17:55
  • $\begingroup$ Can you point to the Wikipedia article? I am inclined to agree that the Conrad's weakening goes too far (i.e. neglects the possibility of torsion elements in $N$). I'm also puzzled why the definition of 'alternating' is restricted to $k\ge 2$, since it is natural (and useful) to extend it to all $k$. $\endgroup$ – qman Mar 19 at 22:59

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