## Tag `05WQ`

Chapter 10: Commutative Algebra > Section 10.150: Henselization and strict henselization

**Henselization is compatible with quotients.**

Lemma 10.150.10. Let $R$ be a local ring with henselization $R^h$. Let $I \subset \mathfrak m_R$. Then $R^h/IR^h$ is the henselization of $R/I$.

Proof.This is a special case of Lemma 10.150.9. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 41142–41150 (see updates for more information).

```
\begin{lemma}
\label{lemma-quotient-henselization}
\begin{slogan}
Henselization is compatible with quotients.
\end{slogan}
Let $R$ be a local ring with henselization $R^h$.
Let $I \subset \mathfrak m_R$.
Then $R^h/IR^h$ is the henselization of $R/I$.
\end{lemma}
\begin{proof}
This is a special case of
Lemma \ref{lemma-quasi-finite-henselization}.
\end{proof}
```

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