Remark 1

— Clearly the annihilator of every element ≠ 0 of a τ-primary module has the form $A\pi^k$ (k > 0 an integer), since it is a principal ideal containing a power of τ. Let x be an element of M; for each $\pi \in P$, let $x_\pi$ be the component of x in $M_\pi$; the annihilator of x is the lcm of the annihilators of the nonzero $x_\pi$, but by the above it is equal in this case to the product of the annihilators of the nonzero $x_\pi$ (VI, p. 16, Prop. 12 (DIV)).