Lemma 15.90.3. Let $R$ be a ring, let $f \in R$, and let $R \to R'$ be a ring map which induces isomorphisms $R/f^ nR \to R'/f^ nR'$ for $n > 0$. The $R$-module $R' \oplus R_ f$ is faithful: for every nonzero $R$-module $M$, the module $M \otimes _ R (R' \oplus R_ f)$ is also nonzero. For example, if $M$ is nonzero, then $M \otimes _ R (R^\wedge \oplus R_ f)$ is nonzero.

Proof. If $M \neq 0$ but $M \otimes _ R R_ f = 0$, then $M$ is $f$-power torsion. By Lemma 15.90.2 we find that $M \otimes _ R R' \cong M \neq 0$. The last statement is a special case of the first statement by Lemma 15.90.1. $\square$

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