Lemma 29.46.4. Let $A \subset B$ be a ring extension. If there exists $b \in B$, $b \not\in A$ and an integer $n \geq 2$ with $b^ n \in A$ and $b^{n + 1} \in A$, then there exists a $b' \in B$, $b' \not\in A$ with $(b')^2 \in A$ and $(b')^3 \in A$.

**Proof.**
Let $b$ and $n$ be as in the lemma. Then all sufficiently large powers of $b$ are in $A$. Namely, $(b^ n)^ k(b^{n + 1})^ i = b^{(k + i)n + i}$ which implies any power $b^ m$ with $m \geq n^2$ is in $A$. Hence if $i \geq 1$ is the largest integer such that $b^ i \not\in A$, then $(b^ i)^2 \in A$ and $(b^ i)^3 \in A$.
$\square$

## 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Your email address will not be published.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: