# Krull–Akizuki theorem

In algebra, the **Krull–Akizuki theorem** states the following: let *A* be a one-dimensional reduced noetherian ring,^{[1]} *K* its total ring of fractions. If *B* is a subring of a finite extension *L* of *K* containing *A* and is not a field, then *B* is a one-dimensional noetherian ring. Furthermore, for every nonzero ideal *I* of *B*, is finite over *A*.^{[2]}

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.

## Proof

Here, we give a proof when . Let be minimal prime ideals of *A*; there are finitely many of them. Let be the field of fractions of and the kernel of the natural map . Then we have:

Now, if the theorem holds when *A* is a domain, then this implies that *B* is a one-dimensional noetherian domain since each is and since . Hence, we reduced the proof to the case *A* is a domain. Let be an ideal and let *a* be a nonzero element in the nonzero ideal . Set . Since is a zero-dim noetherian ring; thus, artinian, there is an *l* such that for all . We claim

Since it suffices to establish the inclusion locally, we may assume *A* is a local ring with the maximal ideal . Let *x* be a nonzero element in *B*. Then, since *A* is noetherian, there is an *n* such that and so . Thus,

Now, assume *n* is a minimum integer such that and the last inclusion holds. If , then we easily see that . But then the above inclusion holds for , contradiction. Hence, we have and this establishes the claim. It now follows:

Hence, has finite length as *A*-module. In particular, the image of *I* there is finitely generated and so *I* is finitely generated. Finally, the above shows that has zero dimension and so *B* has dimension one.

## References

- ↑ In this article, a ring is commutative and has unity.
- ↑ Template:Harvnb