## Tag `02JM`

Chapter 10: Commutative Algebra > Section 10.35: Finite and integral ring extensions

Lemma 10.35.15. Let $A \to B \to C$ be ring maps.

- If $A \to C$ is integral so is $B \to C$.
- If $A \to C$ is finite so is $B \to C$.

Proof.Omitted. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 7386–7393 (see updates for more information).

```
\begin{lemma}
\label{lemma-integral-permanence}
Let $A \to B \to C$ be ring maps.
\begin{enumerate}
\item If $A \to C$ is integral so is $B \to C$.
\item If $A \to C$ is finite so is $B \to C$.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
```

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