In algebra, a Mori domain, named after
Yoshiro Mori by , is an
integral domain
In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibilit ...
satisfying the
ascending chain condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly Ideal (ring theory), ideals in certain commutative rings. These conditions p ...
on integral
divisorial ideals.
Noetherian domains and
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 both have this property. A commutative ring is a Krull domain if and only if it is a Mori domain and
completely integrally closed.
[Bourbaki AC ch. VII ยง1 no. 3 th. 2] A polynomial ring over a Mori domain need not be a Mori domain. Also, the
complete integral closure of a Mori domain need not be a Mori (or, equivalently, Krull) domain.
Notes
References
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Commutative algebra
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