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Result pertaining to division rings
abstract algebra, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to division rings. It says
Cartan–Brauer–Hua_theorem
Nonexistence result for combinatorial block designs
The Bruck–Ryser–Chowla theorem is a result on the combinatorics of symmetric block designs that implies nonexistence of certain kinds of design. It states
Bruck–Ryser–Chowla_theorem
theorem, named after Jacob Levitzki, states that in a right Noetherian ring, every nil one-sided ideal is necessarily nilpotent. Levitzky's theorem is
Levitzky's_theorem
Basic result in the algebraic theory of quadratic forms, on extending isometries
"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory. In mathematics, Witt's theorem, named after
Witt's_theorem
Result in algebra
In mathematics, Wedderburn's little theorem states that every finite division ring is a field; thus, every finite domain is a field. In other words, for
Wedderburn's_little_theorem
Result in ring theory
In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie
Goldie's_theorem
Commutative algebra theorem
The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra concerning the relationship between
Quillen–Suslin_theorem
Mathematic theorem
In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional
Cartan–Dieudonné_theorem
American academic (1913–2002)
until August 1945. Los Angeles: Tomash Publishers. ISBN 978-0-938228-08-0. LAMS-2532. Retrieved February 20, 2014. {{cite book}}: ISBN / Date incompatibility
David_Hawkins_(philosopher)
abstract algebra, in particular ring theory, the Akizuki–Hopkins–Levitzki theorem connects the descending chain condition and ascending chain condition in
Hopkins–Levitzki_theorem
Branch of pure mathematics
understand but are very difficult to solve. Examples of this are Fermat's Last Theorem, which was proved 358 years after the original formulation, and Goldbach's
Number_theory
Chinese-American mathematician (born 1949)
partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors
Shing-Tung_Yau
Mathematical ring with well-behaved ideals
general theorems on rings rely heavily on the Noetherian property (for example, the Lasker–Noether theorem and the Krull intersection theorem). Noetherian
Noetherian_ring
In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is
Kaplansky's theorem on projective modules
Kaplansky's_theorem_on_projective_modules
Division with remainder of integers
ambiguity, Euclidean division. The theorem is frequently referred to as the division algorithm (although it is a theorem and not an algorithm), because its
Euclidean_division
Algebraic structure also called skew field
a b–1 ≠ b–1 a. A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite
Division_ring
Non-associative algebras with positive-definite quadratic form
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz, published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital
Hurwitz's theorem (composition algebras)
Hurwitz's_theorem_(composition_algebras)
Branch of algebra
a theorem which is fundamental for algebraic geometry, and is stated and proved in terms of commutative algebra. Similarly, Fermat's Last Theorem is
Ring_theory
Limitation on the minimum time for a quantum system to evolve between two states
system to evolve between two distinguishable (orthogonal) states. QSL theorems are closely related to time-energy uncertainty relations. In 1945, Leonid
Quantum_speed_limit
Statement in abstract algebra
algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated
Structure theorem for finitely generated modules over a principal ideal domain
Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain
German mathematician (1882–1935)
contributions to abstract algebra. She also proved Noether's first and second theorems, which are fundamental in mathematical physics. Noether was described by
Emmy_Noether
Algebraic structure
Artin–Zorn theorem generalizes the theorem to alternative rings: every finite simple alternative ring is a field. The Artin–Wedderburn theorem is a classification
Noncommutative_ring
square in F and F is Euclidean. This "going-down theorem" may be deduced from the Diller–Dress theorem. A quadratic closure of a field F is a quadratically
Quadratically_closed_field
Historical critique of quantum mechanics
local hidden-variable theory. This second result became known as the Bell theorem. To understand the first result, consider the following toy hidden-variable
Einstein–Podolsky–Rosen paradox
Einstein–Podolsky–Rosen_paradox
Field in which every sum of two squares is a square
line but where the sum of the angles of a triangle is at least π. This theorem states that if E/F is a finite field extension, and E is Pythagorean, then
Pythagorean_field
The 15 and 290 theorems give conditions for a quadratic form to represent all positive integers. Lam (2005) p.10 Rajwade (1993) p.146 Lam (2005) p.36 Serre
Universal_quadratic_form
Ideal in a ring which has properties similar to prime elements
{\displaystyle \mathbb {C} \times \mathbb {C} } (by the Chinese remainder theorem). This shows that the ideal ( x 2 + y 2 − 1 , x ) ⊂ C [ x , y ] {\displaystyle
Prime_ideal
Ring in which every ideal is principal
Chinese Remainder theorem to a minimal primary decomposition of the zero ideal. There is also the following result, due to Hungerford: Theorem (Hungerford):
Principal_ideal_ring
General concept and operation in mathematics
mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a one-to-one fashion, often
Duality_(mathematics)
Problem of constructing equal-area shapes
proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ( π {\displaystyle \pi } ) is a transcendental number
Squaring_the_circle
is 0, 1 or 3. A theorem of Albert states that every central simple algebra of degree 4 and exponent 2 is a biquaternion algebra. Lam 2005, p. 60. Szymiczek
Biquaternion_algebra
Polish mathematician and physicist (1909–1984)
April 1947). "Statistical Methods in Neutron Diffusion" (PDF). LANL report LAMS-551. Retrieved 23 October 2011. Metropolis, Nicholas; Stanislaw Ulam (1949)
Stanisław_Ulam
equations of mathematical physics. In 1896 he gave a proof of the prime number theorem that defines the frequency of prime numbers among the integers (also see
Meanings of minor-planet names: 10001–11000
Meanings_of_minor-planet_names:_10001–11000
バンめし♪ "御伽噺に幕切れを" (Otogibanashi ni Makugire wo) 夜叉姫神楽 from バンめし♪ "Red Cape Theorem" 🔒 メリー・バッド・メルヘン from バンめし♪ "ROOM" Blanc Bunny Bandit from バンめし♪ "逆さま♥シンデレラパレード"
Music of Dance Dance Revolution (2013–present)
Music_of_Dance_Dance_Revolution_(2013–present)
Algebraic structure with addition and multiplication
original on 2019-04-19. Retrieved 2019-04-19. Isaacs (1994), p. 161 Lam (2001), Theorem 3.1 Lang (2005), Ch V, §3. Serre (2006), p. 3 Serre (1979), p. 158
Ring_(mathematics)
Branch of mathematics
of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained
Geometry
Ancient Chinese mathematics text
also the mathematical proof given in the treatise for the Pythagorean theorem. The influence of The Nine Chapters greatly assisted the development of
The Nine Chapters on the Mathematical Art
The_Nine_Chapters_on_the_Mathematical_Art
Geometric concept of a 2D space with "points at infinity" adjoined
such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes. A projective plane is a rank 2 incidence
Projective_plane
(1960). Henry O. Pollak, 98, Austrian-American mathematician (Graham–Pollak theorem, Gilbert–Pollak conjecture). Phil Regan, 89, American baseball player (Los
Deaths_in_2026
n , n ) {\displaystyle \;(1,n,n)\;} is admissible. The Hurwitz–Radon theorem states that ( ρ ( n ) , n , n ) {\displaystyle \;\left(\rho (n),n,n\right)\;}
Hurwitz_problem
Homomorphisms between simple modules over the same ring are isomorphisms or zero
and the properties of the endomorphism ring of M {\displaystyle M} . Theorem (Lam 2001, §19): A module is said to be strongly indecomposable if its endomorphism
Schur's_lemma
Subject area in mathematics
group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of
Algebraic_K-theory
Mathematical treatise
earliest example of the Chinese remainder theorem, a key tool to understanding and resolving Diophantine equations. Lam Lay Yong and An Tian Se. "Fleeting Footsteps"
Sunzi_Suanjing
Chinese-American mathematician and poet
hedge fund manager. Chern's work, most notably the Chern–Gauss–Bonnet theorem, Chern–Simons theory, and Chern classes, are still highly influential in
Shiing-Shen_Chern
(1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001 Lam, T. Y. (1978). Serre's Conjecture. p. 23. v t e
Stably_free_module
Algebraic structure
"zero-locus theorem") is a theorem, first proved by David Hilbert, which extends to the multivariate case some aspects of the fundamental theorem of algebra
Polynomial_ring
Brothers Grimm (2005), The Imaginarium of Doctor Parnassus (2009), The Zero Theorem (2013) John Gilling Sid James Escape by Night (1953), Interpol (1957),
List of film director and actor collaborations
List_of_film_director_and_actor_collaborations
Abstract algebra concept
a module that is not a direct sum of two nonzero submodules. Azumaya's theorem states that if a module has an decomposition into modules with local endomorphism
Decomposition_of_a_module
Mathematical property
or V (these are also called irreducible representations). Now Maschke's theorem says that any finite-dimensional representation of a finite group is a
Semi-simplicity
Motion of a curve based on its curvature
shortening on a sphere can be used as part of a proof of the tennis ball theorem. This theorem states that every smooth simple closed curve on the sphere that
Curve-shortening_flow
Branch of mathematics that studies abstract algebraic structures
over fields of characteristic zero) include finite groups (see Maschke's theorem), compact groups, and semisimple Lie algebras. In cases where complete
Representation_theory
Algebra over a field with only invertible elements and zero
numbers that are finite-dimensional as a vector space over R). The Frobenius theorem states that up to isomorphism there are three such algebras: the reals
Division_algebra
Local ring in commutative algebra
structure theorem to case of codimension 4. Eisenbud (1995), pg 525. Eisenbud (1995), Proposition 21.5. Huneke (1999), Theorem 9.1. Lam (1999), Theorems 3.15
Gorenstein_ring
Mathematical ring whose elements are matrices
Clifford algebra Hurwitz's theorem (normed division algebras) Generic matrix ring Sylvester's law of inertia Lam (1999), Theorem 3.1 Lam (2001). Lang (2005)
Matrix_ring
Application of mathematical and statistical methods in finance
Financial modeling; Asset pricing. The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes
Mathematical_finance
Array of numbers
III.2.1. Brown (1991), Theorem III.2.12. Brown (1991), Corollary III.2.16. Mirsky (1990), Theorem 1.4.1. Brown (1991), Theorem III.3.18. Eigen means "own"
Matrix_(mathematics)
Direct sum of irreducible modules
of Artinian semisimple rings is well understood by the Artin–Wedderburn theorem, which exhibits these rings as finite direct products of matrix rings.
Semisimple_module
Rule system for formal languages
Ullman 1979, p. 131-132, Theorem 6.2. Hopcroft & Ullman 1979, p. 132-134, Theorem 6.3. Hopcroft & Ullman 1979, p. 135-136, Theorem 6.5. Hopcroft & Ullman
Context-free_grammar
Abelian group in which every element can, in some sense, be divided by positive integers
classification of countable reduced periodic abelian groups is given by Ulm's theorem. Several distinct definitions generalize divisible groups to divisible
Divisible_group
Type of algebras, possibly non associative
real algebras with positive definite forms was delimited by the Hurwitz's theorem (composition algebras). In 1931 Max Zorn introduced a gamma (γ) into the
Composition_algebra
Prototype of Secure Sockets Layer
Modules I — Composition Theorem". IEEE Transactions on Software Engineering. 20: 55–71. doi:10.1109/32.263755. Retrieved 21 July 2019. Lam, Simon; Shankar, Udaya;
Secure_Network_Programming
Number which describes the structure of the set of squares in a given field
two squares, so p = 2 {\displaystyle p=2} . By Lagrange's four-square theorem, every positive rational number is a sum of four squares, and not all are
Pythagoras_number
module. (Lam 1999) A standard result is that a right Noetherian domain is a right Ore domain. In fact, we can recover this result from another theorem attributed
Uniform_module
American mathematician
theory to the field, and contains the first complete proof of the Markov theorem on braids. In 1973, she joined the faculty at Barnard College, where she
Joan_Birman
Generalizations of prime ideals and prime rings
ascending chain condition on right annihilators of its subsets. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have
Semiprime_ring
invariants and the signatures coming from real embeddings. Hasse–Minkowski theorem Lam (2005) p.118 Milnor & Husemoller (1973) p.79 Serre (1973) p.36 Serre
Hasse invariant of a quadratic form
Hasse_invariant_of_a_quadratic_form
Algebraic structure
p. 148, Theorem 2.23. Fraleigh & Katz (1967), p. 368, Theorem 7.2 Hazewinkel, Gubareni & Kirichenko (2004), p.166, Theorem 7.2.1. T. Y. Lam; Manuel L
Principal_ideal_domain
Topics referred to by the same term
(linguistics), a level of syntactic representation Logical framework, in automated theorem proving LF (logical framework), a particular logical framework Missile
LF
Linear operator
that both s and n are polynomials in x. Lam (2001), p. 39 Jacobson 1979, A paragraph before Ch. II, § 5, Theorem 11. This is trivial by the definition in
Semisimple_operator
Pythagorean theorem. Theorems on the lengths of chords are essentially applications of the modern law of sines. We have seen that Archimedes' theorem on the
Timeline of scientific discoveries
Timeline_of_scientific_discoveries
Nothing Bad Can Happen Our Heroes Died Tonight Patrick She Wolf The Zero Theorem U.S. premieres Afflicted Almost Human Blue Ruin Borgman Commando: A One
List of Fantastic Fest editions
List_of_Fantastic_Fest_editions
Physical principle that only immediate surroundings can influence an object
physically separated measurements. In 1964, John Stewart Bell formulated Bell's theorem, an inequality which, if violated in actual experiments, implies that quantum
Principle_of_locality
Mathematical object in abstract algebra
module is injective if and only if the ring is hereditary, (Lam 1999, Th. 3.22). Bass-Papp Theorem states that every infinite direct sum of right (left) injective
Injective_module
French mathematician (1920–1996)
his peers and from critics of his stance on evolution. Several notable theorems and objects in mathematics as well as computer science bear his name (for
Marcel-Paul_Schützenberger
Constant equal to twice pi
Euler's identity, eiπ + 1 = 0, sometimes claimed to be "the most beautiful theorem in mathematics" is made less elegant rendered as eiτ/2 + 1 = 0. Hartl has
Tau_(mathematics)
American mathematician
Feit–Thompson theorem McKay–Thompson series Quadratic pair Thompson factorization Thompson order formula Thompson subgroup Thompson transitivity theorem Thompson
John_G._Thompson
scientist and translator Kenneth Appel (1932–2013), proved four-color theorem Zvi Arad (1942–2018), mathematician Vladimir Arnold (1937–2010), mathematician;
List_of_Jewish_mathematicians
Quadratic form for which there is a non-zero vector on which the form evaluates to zero
isotropic forms are usually much easier to handle. By Witt's decomposition theorem, every inner product space over a field is an orthogonal direct sum of
Isotropic_quadratic_form
nonsingular is equivalent to being a reduced ring. Johnson's Theorem (due to R. E. Johnson (Lam 1999, p. 376)) contains several important equivalences. For
Singular_submodule
products of primitive rings, which are described by the Jacobson density theorem. A ring is called semiprimitive or Jacobson semisimple if its Jacobson
Semiprimitive_ring
Use of mathematical and statistical methods in finance
securities, laying the groundwork for the development of the fundamental theorem of asset pricing. The various short-rate models (beginning with Vasicek
Quantitative analysis (finance)
Quantitative_analysis_(finance)
Polynomial with all terms of degree two
over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed
Quadratic_form
Mathematical problem
MacNeish's theorem does not give a very good lower bound, for instance if n ≡ 2 (mod 4), that is, there is a single 2 in the prime factorization, the theorem gives
Mutually orthogonal Latin squares
Mutually_orthogonal_Latin_squares
Deviations from local realism
the many-worlds interpretation which violates an assumption of Bell's theorem. Quantum nonlocality does not allow for faster-than-light communication
Quantum_nonlocality
Cradle of civilization in North Africa
shown on the right. Ancient Egyptian mathematicians knew the Pythagorean theorem as an empirical formula. They were aware, for example, that a triangle
Ancient_Egypt
Submodule of a mathematical ring
ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization
Ideal_(ring_theory)
Ordered field where every nonnegative element is a square
E is Euclidean, then so is F. This "going-down theorem" is a consequence of the Diller–Dress theorem. The real constructible numbers, those (signed)
Euclidean_ordered_field
(Mathematical) ring with a unique maximal ideal
}m^{i}=\{0\}} (Krull's intersection theorem), and it follows that R with the m-adic topology is a Hausdorff space. The theorem is a consequence of the Artin–Rees
Local_ring
Multivalued function in mathematics
{\displaystyle W_{k}(z)} is algebraic. Then by the Lindemann–Weierstrass theorem we have e W k ( z ) {\displaystyle e^{W_{k}(z)}} is transcendental, but
Lambert_W_function
Classification in abstract algebra
even, the algebra Cln(C) is central simple and so by the Artin–Wedderburn theorem is isomorphic to a matrix algebra over C. When n is odd, the center includes
Classification of Clifford algebras
Classification_of_Clifford_algebras
American mathematician
combinatorics in the 20th century. He is the namesake of the Bruck–Ryser–Chowla theorem, Ryser's formula for the computation of the permanent of a matrix, and
H._J._Ryser
East Asian ethnic group
(published 2020): 49. doi:10.1186/s41065-020-00162-w. PMC 7724877. PMID 33292737. Lams, Lutgard (September 2013). "Critical Han studies: the history, representation
Han_Chinese
multiple c = au = bv with u, v not zero divisors. In this case, Ore's theorem guarantees the existence of an over-ring called the (right or left) classical
Ore_condition
that free modules over an IBN ring satisfy an analogue of the dimension theorem for vector spaces: any two bases for a free module over an IBN ring have
Invariant_basis_number
Ball used in cue sports
motion. Idealized, frictionless billiard balls are a staple of mathematical theorems and physics models, and figure in dynamical billiards, scattering theory
Billiard_ball
Finite dimensional algebra over a field whose central elements are that field
F as it has infinite dimension as a F-module. By the Artin–Wedderburn theorem, a finite-dimensional simple algebra A is isomorphic to the matrix algebra
Central_simple_algebra
Day of the year
Investigation). 1918 – Emmy Noether's paper, which became known as Noether's theorem was presented at Göttingen, Germany, from which conservation laws are deduced
July_26
Type of ring in non-commutative algebra
the Wedderburn-Artin theorem" (PDF). New Zealand J. Math. 22: 83–86. Henderson, D. W. (1965). "A short proof of Wedderburn's theorem". Amer. Math. Monthly
Simple_ring
Ring without nonzero zero divisors
finite domain is automatically a finite field, by Wedderburn's little theorem. The quaternions form a noncommutative domain. More generally, any division
Domain_(ring_theory)
over an algebraically closed field has u ≤ 2; this follows from Tsen's theorem that such a field is quasi-algebraically closed. If F is a non-real global
U-invariant
LAMS THEOREM
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