In algebra, the Mori–Nagata theorem introduced by and , states the following: let ''A'' be a
noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
reduced commutative ring with the
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 embeds ...
''K''. Then the
integral closure of ''A'' in ''K'' is a
direct product of ''r''
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 ...
s, where ''r'' is the number of
minimal prime ideal In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes.
Definitio ...
s of ''A''.
The theorem is a partial generalization of the
Krull–Akizuki theorem, which concerns a one-dimensional noetherian domain. A consequence of the theorem is that if ''R'' is a
Nagata ring In commutative algebra, an N-1 ring is an integral domain A whose integral closure in its quotient field is a finitely generated A-module. It is called a Japanese ring (or an N-2 ring) if for every finite extension L of its quotient field K, the ...
, then every ''R''-subalgebra of finite type is again a Nagata ring .
The Mori–Nagata theorem follows from
Matijevic's theorem.
References
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Commutative algebra
Theorems in ring theory
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