## A fact about simple alternating tensors

Let $R$ be a commutative ring with 1 and let $M$ be an $(R,R)$-bimodule such that $rm = mr$ for all $r \in R$ and $m \in M$. Prove that $m \wedge n_1 \wedge \cdots \wedge n_k = (-1)^k (n_1 \wedge \cdots \wedge n_k \wedge m)$.

We proceed by induction on $k$. The result is clear if $k$ is 0 or 1. Suppose the result holds for some $k$. Now $m \wedge n_1 \wedge \cdots \wedge n_k \wedge n_{k+1} = (m \wedge n_1 \wedge \cdots \wedge n_k) \wedge n_{k+1}$ $= (-1)^k(n_1 \wedge \cdots n_k \wedge m) \wedge n_{k+1}$ $= (-1)^{k+1}(n_1 \wedge \cdots \wedge n_{k+1} \wedge m$.