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.

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