Lemma 10.31.3. Let $R \to R'$ be a ring map and let $I \subset R$ be a locally nilpotent ideal. Then $IR'$ is a locally nilpotent ideal of $R'$.

Proof. This follows from the fact that if $x, y \in R'$ are nilpotent, then $x + y$ is nilpotent too. Namely, if $x^ n = 0$ and $y^ m = 0$, then $(x + y)^{n + m - 1} = 0$. $\square$

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