The Stacks project

Here is a very minor comment: in the proof of Krull-Akizuki, you should assume $I$ is nonzero, or else the "Clearly ..." statement is false.

Thanks, fixed here.

An alternative way to see that $I\cap R\ne(0)$ is as follows: Take a non-zero $x\in I$. Since $L/K$ is finite, we can take the minimal polynomial $p$ of $x$ over $K$. Write it as $a_nx^n+\cdots+a_0=0$ with $a_i\in K$. By multiplying by some elements in $R$ if necessary, we may assume $a_i\in R,\,\forall i$. Also $a_0\ne0$ as $p$ is minimal. Now $a_nx^n+\cdots+a_1x\in I$, so $a_0\in I\cap R$.

In the proof of Krull-Akizuki the beggining seems to be circular, or not well-written: To begin we may assume that $L$ is the fraction field of $A$ by replacing $L$ by the fraction field of $A$ if necessary.