Proposition 2

— Let $A$ be a commutative ring, and let $E$ be a direct sum of $n$ cyclic modules $A / a_k$, where the $a_k$ satisfy $a_1 \subset a_2 \subset \ldots \subset a_n$. Then, for $1 \leq p \leq n$, the ideal $a_p$ is the annihilator of $\bigwedge^p E$; if $a_n \neq A$ then $\bigwedge^p E \neq 0$ for $1 \leq p \leq n$ and $\bigwedge^m E = 0$ for $m > n$.

Indeed, in the notation of Prop. 1, we have $a_p = a_{s(H)}$, where $s(H)$ denotes the greatest element of the subset $H$. Since $s(H) \geq p$ for every $p$-element subset $H$, and since $s(H) = p$ for $H = \{ 1, \ldots, p \}$, it follows that $a_p$ is the intersection of $a_r$, as $H$ varies over the set of $p$-element subsets of $(1, n)$; the ideal $a_p$ is thus indeed the annihilator of $\bigwedge^p E$, by Prop. 1.