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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: