Add a sentence to a proof

THanks to Et
https://stacks.math.columbia.edu/tag/00J8#comment-8248

algebra.tex

@@ -12599,9 +12599,11 @@ \section{Artinian rings}
 Set $J = \{ x\in R \mid xI^n = 0\}$. We have to show $J = R$.
 If not, choose an ideal $J' \not = J$, $J \subset J'$ minimal (possible
 by the Artinian property). Then $J' = J + Rx$ for some $x \in R$.
-By NAK, Lemma \ref{lemma-NAK}, we have $IJ' \subset J$.
+By minimality we have $J + IJ' = J$ or $J + IJ' = J'$. In the latter
+case we get $J' = J + Ix$ and by Lemma \ref{lemma-NAK}
+we obtain $J = J'$ a contradiction.
 Hence $xI^{n + 1} \subset xI \cdot I^n \subset J \cdot I^n = 0$.
-Since $I^{n + 1} = I^n$ we conclude $x\in J$. Contradiction.
+Since $I^{n + 1} = I^n$ we conclude $x \in J$. Contradiction.
 \end{proof}
 
 \begin{lemma}