In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
, the Krull–Akizuki theorem states the following: Let ''A'' be a
one-dimensional reduced noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
, ''K'' its
total ring of fractions
In abstract algebra, the total quotient ring or total ring of fractions is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embedding ...
. Suppose ''L'' is a finite extension of ''K''. If
and ''B'' is reduced,
then ''B'' is a noetherian ring of dimension at most one. Furthermore, for every nonzero ideal
of ''B'',
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
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over a subring ''A'' of ''B'' if ''b'' is a root of some monic polynomial over ''A''.
If ''A'', ''B'' are fields, then the notions of "integral over" and ...
of a
Dedekind domain
In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...
''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 In algebra, the Mori–Nagata theorem introduced by and , states the following: let ''A'' be a noetherian ring, noetherian reduced ring, reduced commutative ring with the total ring of fractions ''K''. Then the integral closure of ''A'' in ''K'' is ...
states that the integral closure of a noetherian domain is a
Krull domain In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimensional generalization of Dedekind domains, which a ...
.
Proof
First observe that
and ''KB'' is a finite extension of ''K'', so we may assume without loss of generality that
.
Then
for some
.
Since each
is integral over ''K'', there exists
such that
is integral over ''A''.
Let