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Krull's and theorem
* Krull's theorem can fail for rings without unity.
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.
# REDIRECT Krull's principal ideal theorem
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.
# REDIRECT Krull's principal ideal theorem
** Krull's intersection theorem
** Krull's principal ideal theorem
** Krull's theorem
* Krull's intersection theorem
* Krull's principal ideal theorem
* Krull's 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.

theorem and 1929
The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 ( and a rewritten version of the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.
* Siegel's theorem on integral points, from 1929
Vinogradov ), a related 1929 theorem ( of Knaster, Borsuk and Mazurkiewicz ) has also become known as the Sperner lemma-this point is discussed in the English translation ( ed.
He is now best known for his contribution to the principal ideal theorem in the form of his Beweis des Hauptidealsatzes für Klassenkörper algebraischer Zahlkörper ( 1929 ).

theorem and ):
" It also contains the general statement of the Pythagorean theorem ( for the sides of a rectangle ): " The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together.
Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary group U ( t ): H → H such that
An important example is the ring Z / nZ of integers modulo n. If n is written as a product of prime powers ( see fundamental theorem of arithmetic ):
Observe that and hence by the argument above we may apply Fubini's theorem again ( i. e. interchange the order of integration ):
From the last equation, we can deduce Goldbach's theorem ( named after Christian Goldbach ): no two Fermat numbers share a common factor.
:; Eigendecomposition of a symmetric matrix ( Decomposition according to Spectral theorem ): S = QΛQ < sup > T </ sup >, S symmetric, Q orthogonal, Λ diagonal.
The product of normal operators that commute is again normal ; this is nontrivial and follows from Fuglede's theorem, which states ( in a form generalized by Putnam ):
Use the binomial theorem to expand ( a + b )< sup > n + m − 1 </ sup > ( with commutativity assumed ):
Gauss presented the theorem in this way ( translated from Latin ):
The first part of the Peter – Weyl theorem asserts (; ):
Consider the following theorem ( which is a case of the Pigeonhole Principle ):
To derive Green's theorem, begin with the divergence theorem ( otherwise known as Gauss's theorem ):
The first significant result in what later became model theory was Löwenheim's theorem in Leopold Löwenheim's publication " Über Möglichkeiten im Relativkalkül " ( 1915 ):
A corollary to the theorem is then the primitive element theorem in the more traditional sense ( where separability was usually tacitly assumed ):
which leads to the generalised least squares version of the Gauss-Markov theorem ( Chiles & Delfiner 1999, p. 159 ):
More precisely, their theorem states that there is no apportionment system that has the following properties ( as the example we take the division of seats between parties in a system of proportional representation ):
If only a time series is available, the phase space can be reconstructed by using a time delay embedding ( see Takens ' theorem ):
However, the theorem does not rely upon the axiom of choice in the separable case ( see below ): in this case one actually has a constructive proof.
Tarski's undefinability theorem ( general form ): Let ( L, N ) be any interpreted formal language which includes negation and has a Gödel numbering g ( x ) such that for every L-formula A ( x ) there is a formula B such that B ↔ A ( g ( B )) holds.
The classical prime number theorem gives an asymptotic formula for π ( x ):
Precisely, the s-cobordism theorem ( the s stands for simple-homotopy equivalence ), proved independently by Barry Mazur, John Stallings, and Dennis Barden, states ( assumptions as above but where M and N need not be simply connected ):

theorem and Every
** Well-ordering theorem: Every set can be well-ordered.
* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).
* Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ".
* Every pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.
Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.
In cybernetics, the Good Regulator or Conant-Ashby theorem is stated " Every Good Regulator of a system must be a model of that system ".
Every Tychonoff cube is compact Hausdorff as a consequence of Tychonoff's theorem.
Every proper rotation is the composition of two reflections, a special case of the Cartan – Dieudonné theorem.
Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R < sup > 3 </ sup > which is called the axis of rotation ( this is Euler's rotation theorem ).
Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies.
Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection J from the definition is bijective, by the rank-nullity theorem.
Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof.
Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by
For example, to study the theoremEvery bounded sequence of real numbers has a supremum ” it is necessary to use a base system which can speak of real numbers and sequences of real numbers.
# Every K3 surface is Kähler ( by a theorem of Y .- T. Siu ).
* Every finite-dimensional simple algebra over R must be a matrix ring over R, C, or H. Every central simple algebra over R must be a matrix ring over R or H. These results follow from the Frobenius theorem.
* Every automorphism of a central simple algebra is an inner automorphism ( follows from Skolem – Noether theorem ).
* Conway's cosmological theorem: Every sequence eventually splits into a sequence of " atomic elements ", which are finite subsequences that never again interact with their neighbors.
Every comparability graph is perfect: this is essentially just Mirsky's theorem, restated in graph-theoretic terms.
Every rotation in three dimensions is defined by its axis — a direction that is left fixed by the rotation — and its angle — the amount of rotation about that axis ( Euler rotation theorem ).
Every classical theorem that applies to the natural numbers applies to the non-standard natural numbers.
* Moise's theoremEvery 3-manifold has a triangulation, unique up to common subdivision

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