Krull–Akizuki theorem

Remarks

The general case reduces to the case $L = K$. There the key step is the bound $\ell_A(B/aB) \le \ell_A(A/aA)$ for each nonzero $a \in A$, where $\ell_A$ is the length; the right side is finite because $A$ has dimension $1$. Each nonzero ideal $J$ of $B$ contains a nonzero element $a$ of $A$, so $J/aB$ has finite length, and $J$ is finitely generated. A main application: the integral closure of $A$ in $L$ is a Noetherian integrally closed domain of dimension at most $1$, that is, a Dedekind domain or a field. For example, with $A = \Z$ and $L$ a number field, this shows that the ring of integers of $L$ is Noetherian. The ring $B$ need not be a finitely generated $A$-module, so the result is not a consequence of Hilbert's basis theorem.

Going deeper

For a student

The theorem is the standard route to the following fact. The integral closure of a one-dimensional Noetherian domain in a finite extension of its fraction field is a Dedekind domain, even when it is not a finite module. It is special to dimension $1$: for any Noetherian domain, the Mori–Nagata theorem only gives a Krull domain, and the integral closure can fail to be Noetherian. The proof uses lengths of modules and the fact that $A/aA$ is Artinian for $a \ne 0$.