Lemma 47.2.4. Let $R$ be a ring. Let $M \subset N$ be $R$-modules. The following are equivalent

$M \subset N$ is an essential extension,

for all $x \in N$ nonzero there exists an $f \in R$ such that $fx \in M$ and $fx \not= 0$.

Lemma 47.2.4. Let $R$ be a ring. Let $M \subset N$ be $R$-modules. The following are equivalent

$M \subset N$ is an essential extension,

for all $x \in N$ nonzero there exists an $f \in R$ such that $fx \in M$ and $fx \not= 0$.

**Proof.**
Assume (1) and let $x \in N$ be a nonzero element. By (1) we have $Rx \cap M \not= 0$. This implies (2).

Assume (2). Let $N' \subset N$ be a nonzero submodule. Pick $x \in N'$ nonzero. By (2) we can find $f \in R$ with $fx \in M$ and $fx \not= 0$. Thus $N' \cap M \not= 0$. $\square$

Your email address will not be published.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: