Warning 8.4.11. The finite-coproduct step above is the one place where some care is needed: given \(x_1 \in \pi _n(M_1)\) annihilated by \(s_1\) and \(x_2 \in \pi _n(M_2)\) annihilated by \(s_2\), the product \(s_2s_1\) need not annihilate \((x_1,x_2)\), since \(s_2s_1 \cdot x_2 = s_2\cdot (s_1 \cdot x_2)\) and \(s_1 \cdot x_2\) has no reason to vanish. What saves the argument is that we may instead first apply \(s_2\), killing the second coordinate, and then use the \(S\)-nilpotence of \(M_1\) to annihilate the element \(s_2 \cdot x_1\) (rather than \(x_1\) itself). No commutativity or Ore condition is needed.
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