In algebra, a Mori domain, named after
Yoshiro Mori by , is an
integral domain
In mathematics, specifically abstract algebra, 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 set ...
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 ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These con ...
on integral
divisorial ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral dom ...
s.
Noetherian domain
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-Noether ...
s 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 §1 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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*{{Citation , last1=Querré , first1=J. , title=Cours d'algèbre , url=https://books.google.com/books/about/Cours_d_alg%C3%A8bre.html?id=X1LQAAAAMAAJ , publisher=Masson , location=Paris , mr=0465632 , year=1976, isbn=9782225441875
Commutative algebra