In algebra, the **Krull–Akizuki theorem** states the following: let *A* be a one-dimensional reduced noetherian ring, *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*,
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.

Here, we give a proof when
L
=
K
. Let
p
i
be minimal prime ideals of *A*; there are finitely many of them. Let
K
i
be the field of fractions of
A
/
p
i
and
I
i
the kernel of the natural map
B
→
K
→
K
i
. Then we have:

A
/
p
i
⊂
B
/
I
i
⊂
K
i
.

Now, if the theorem holds when *A* is a domain, then this implies that *B* is a one-dimensional noetherian domain since each
B
/
I
i
is and since
B
=
∏
B
/
I
i
. Hence, we reduced the proof to the case *A* is a domain. Let
0
≠
I
⊂
B
be an ideal and let *a* be a nonzero element in the nonzero ideal
I
∩
A
. Set
I
n
=
a
n
B
∩
A
+
a
A
. Since
A
/
a
A
is a zero-dim noetherian ring; thus, artinian, there is an *l* such that
I
n
=
I
l
for all
n
≥
l
. We claim

a
l
B
⊂
a
l
+
1
B
+
A
.
Since it suffices to establish the inclusion locally, we may assume *A* is a local ring with the maximal ideal
m
. Let *x* be a nonzero element in *B*. Then, since *A* is noetherian, there is an *n* such that
m
n
+
1
⊂
x
−
1
A
and so
a
n
+
1
x
∈
a
n
+
1
B
∩
A
⊂
I
n
+
2
. Thus,

a
n
x
∈
a
n
+
1
B
∩
A
+
A
.
Now, assume *n* is a minimum integer such that
n
≥
l
and the last inclusion holds. If
n
>
l
, then we easily see that
a
n
x
∈
I
n
+
1
. But then the above inclusion holds for
n
−
1
, contradiction. Hence, we have
n
=
l
and this establishes the claim. It now follows:

B
/
a
B
≃
a
l
B
/
a
l
+
1
B
⊂
(
a
l
+
1
B
+
A
)
/
a
l
+
1
B
≃
A
/
a
l
+
1
B
∩
A
.

Hence,
B
/
a
B
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
B
/
a
B
has zero dimension and so *B* has dimension one.
◻