Krull–Akizuki theorem

In commutative algebra, the Krull–Akizuki theorem states the following: Let A be a one-dimensional reduced noetherian ring, K its total ring of fractions. Suppose L is a finite extension of K. If ${\displaystyle A\subset B\subset L}$ and B is reduced, then B is a noetherian ring of dimension at most one. Furthermore, for every nonzero ideal ${\displaystyle I}$ of B, ${\displaystyle B/I}$ is finite over A.

Note that the theorem does not say that B is finite over A. The theorem does not extend to higher dimension. One important consequence of the theorem is that the integral closure of a Dedekind domain A in a finite extension of the field of fractions of A is again a Dedekind domain. This consequence does generalize to a higher dimension: the Mori–Nagata theorem states that the integral closure of a noetherian domain is a Krull domain.