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** Krull's theorem
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** and Krull's
** and theorem
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** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
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Krull's and theorem
Krull's principal ideal theorem states that every principal ideal in a commutative Noetherian ring has height one ; that is, every principal ideal is contained in a prime ideal minimal amongst nonzero prime ideals.
The proof of the general case requires some commutative algebra, namely the fact, that the Krull dimension of k is one — see Krull's principal ideal theorem ).
In a Noetherian ring, Krull's height theorem says that the height of an ideal generated by n elements is no greater than n.
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull ( 1899 – 1971 ), gives a bound on the height of a principal ideal in a Noetherian ring.
This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem.
Then its coefficients generate a proper ideal I, which by Krull's theorem ( which depends on the axiom of choice ) is contained in a maximal ideal m of R. Then R / m is a field, and ( R / m ) is therefore an integral domain.
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