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* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.
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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.
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the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.
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 ):
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.
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
* 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 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 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 theorem “ Every 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 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.