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* Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra ; in fact, it is either a two-by-two matrix algebra, or a division algebra.
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Every and central
Every telephone company, whether large or small, determines its own ANAC for each individual central office, which tends to perpetuate the current situation of a mess of overlapping and / or spotty areas of coverage.
Every year the central business district ( with corners at the Municipal Building, Grand Street Fire House and Croton-Harmon High School ) is closed to automobile traffic for music, American food, local fund raisers, traveling, and local artists.
* 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 finite-dimensional simple algebra over C must be a matrix ring over C and hence every central simple algebra over C must be a matrix ring over C.
* Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field.
* Every automorphism of a central simple algebra is an inner automorphism ( follows from Skolem – Noether theorem ).
Dual citizenship was established ( Article 5 ( 3 ): " Every Czechoslovak citizen is at the same time a citizen of the Czech Socialist Republic or the Slovak Socialist Republic "), and many of the former functions of the central government were instead placed under the jurisdiction of the two national governments.
Every telephone company, whether large or small, determines its own ringback numbers for each individual central office, which tends to perpetuate the current situation of a mess of overlapping and / or spotty areas of coverage.
Every summer, a tourist train connects the ancient parts of the city with museums, the central railway station and the passenger ship terminal of Krems.
Every year, at the end of January, the offerings of the members of the society all over the world are forwarded to these central bureaux, and the total amount is divided among all the missions of the earth.
* Retzai Every the Orthodox Amidah prayer, the central prayer of Jewish services, contains the paragraph: " Be favorable, Oh Lord our God, to your people Israel and their prayer, and restore the service of the Holy of Holies of Your House, and accept the fire-offerings of Israel and their prayer with love and favor, and may the service of your people Israel always be favored.
* County Party: Every two years, between the primary election and 2 weeks after the general election, all the precinct committee people-the county party " central Committee "-hold a meeting and elect officers for the county party, a chair, vice chair, secretary and treasurer.
Every connected topological group has a unique universal cover as a topological space, which has a unique group structure as a central extension by the fundamental group.
Every year, thousands of central police and state police personnel are deployed to provide security to pilgrims from potential terror threats.
Every glucosinolate contains a central carbon atom, which is bound via a sulfur atom to the thioglucose group ( making a sulfated ketoxime ) and via a nitrogen atom to a sulfate group.
In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars (= tensoring with a field extension ), i. e. for a suitable field extension K of F, is isomorphic to the 2 × 2 matrix algebra over K.
Every and simple
# Every simple path from a given node to any of its descendant leaves contains the same number of black nodes.
Every simple R-module is isomorphic to a quotient R / m where m is a maximal right ideal of R. By the above paragraph, any quotient R / m is a simple module.
" I might easily have written this story in the traditional manner [...] Every novelist knows the recipe [...] It is not very difficult to follow a simple, chronological scheme which the critics will understand [...] But I, after all, am trying to tell the story of this Chapelizod family in a new way.
Every October, Moriarty plays host to the Pinto Bean Fiesta, which is composed of a bunch of simple games in Crossly Park, as well as a parade and crowning of a " Pinto Bean Queen.
Every one of the infinitely many vertices of G can be reached from v < sub > 1 </ sub > with a simple path, and each such path must start with one of the finitely many vertices adjacent to v < sub > 1 </ sub >.
Every hour that Napoleon could have attacked earlier as he did, would have been is his favour, but the French could not attack in the morning for the simple reason that the entire army had not yet taken its battle positions.
Every closed curve c on X is homologous to for some simple closed curves c < sub > i </ sub >, that is,
# Personal right: Every person has a right to life but this right is restricted and has attached certain duties – simple living is essential.
Every Sámi settlement had its seita, which had no regular shape, and might consist of smooth or odd-looking stones picked out of a stream, of a small pile of stones, of a tree-stump, or of a simple post.
Every and algebra
Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions.
Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.
* Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions.
* Every unital real Banach algebra with no zero divisors, and in which every principal ideal is closed, is isomorphic to the reals, the complexes, or the quaternions.
* Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
* Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional.
Every continuous map f: X → Y induces an algebra homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ in C ( Y ).
Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.
Every associative algebra is obviously power-associative, but so are all other alternative algebras ( like the octonions, which are non-associative ) and even some non-alternative algebras like the sedenions.
Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.
Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.
Every Heyting algebra with exactly one coatom is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new top.