Example 42.68.20. Let $R = \mathbf{Z}_ p$ and let $M = \mathbf{Z}_ p/(p^ l)$. Let $\varphi = p^ b u$ and $\varphi = p^ a v$ with $a, b \geq 0$, $a + b = l$ and $u, v \in \mathbf{Z}_ p^*$. Then a computation as in Example 42.68.19 shows that

with $\alpha = p^ bu, \beta = p^ av \in \mathbf{Z}_ p$. See Lemma 42.68.37 for a more general case (and a proof).

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