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.[1] They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.
In this article, a ring is commutative and has unity.
algebra, a Krullring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931...
the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension...
regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension...
profinite group Krull's intersection, a theorem within algebraic ring theory that describes the behaviors of certain local ringsKrull (film), a 1983 heroic...
rings was introduced by Wolfgang Krull in 1938 under the name Stellenringe. The English term local ring is due to Zariski. A ring R is a local ring if...
Jacobson ring is one in which every prime ideal is an intersection of maximal ideals. Jacobson rings were introduced independently by Wolfgang Krull (1951...
the Krull dimension of a ring, first for Noetherian rings before moving on to expand his theory to cover general valuation rings and Krullrings. To this...
property (for example, the Lasker–Noether theorem and the Krull intersection theorem). Noetherian rings are named after Emmy Noether, but the importance of...
theorem characterizes the automorphisms of simple rings In this section, R denotes a commutative ring. The Krull dimension of R is the supremum of the lengths...
of Krull that an integral domain is a valuation ring if and only if it is a local Bézout domain. It also follows from this that a valuation ring is Noetherian...
Every nonzero ring contains at least one prime ideal (in fact it contains at least one maximal ideal), which is a direct consequence of Krull's theorem. More...
Krullring A Krullring (or Krull domain) is a ring with a well behaved theory of prime factorization. Krull dimension See dimension. Laskerian ring A...
semisimple ring, where nil(A) is the nilradical of A.[citation needed] Every finitely generated module over A has finite length. (see above) A has Krull dimension...
principal. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.) A is a Krull domain and every...
mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set...
of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is an integral domain or a local ring. There...
is the empty scheme. The Krull dimension of the zero ring is −∞. The zero ring is semisimple but not simple. The zero ring is not a central simple algebra...
properties are equivalent: being a semisimple ring; being artinian and reduced; being a reduced Noetherian ring of Krull dimension 0; and being isomorphic to a...
structure of complete Noetherian local rings. Some consequences of Cohen's structure theorem include three conjectures of Krull: Any complete regular equicharacteristic...
to verify (DD4). A Krull domain is a higher-dimensional analog of a Dedekind domain: a Dedekind domain that is not a field is a Krull domain of dimension...
In mathematics, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions...
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