Proposition 6

If M is a finitely generated A-module, $\bigwedge(M)$ is a finitely generated A-module; also, if M admits a generating system with n elements, then $\bigwedge^p(M) = \{0\}$ for $p > n$.

Let $(x_i)_{1 \leq i \leq n}$ be a generating system of M. Every element of $\bigwedge^p(M)$ is a linear combination of elements of the form

where the indices $i_k$ belong to $\{1, n\}$; by Proposition 5, these indices can be assumed to be distinct (otherwise the corresponding element is zero). If $p > n$, there is no such sequence of indices and hence $\bigwedge^p(M) = \{0\}$. If $p \leq n$, these sequences are finite in number, which completes the proof.