[permalink] [id link]

* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.

from
Wikipedia

## Some Related Sentences

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the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure**.**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

**.**

0.636 seconds.