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Page "Artin–Wedderburn theorem" ¶ 6
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Every and finite-dimensional
* Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional.
Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.
Every finite-dimensional vector space is isomorphic to its dual space, but this isomorphism relies on an arbitrary choice of isomorphism ( for example, via choosing a basis and then taking the isomorphism sending this basis to the corresponding dual basis ).
Every Hermitian matrix is a normal matrix, and the finite-dimensional spectral theorem applies.
Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram – Schmidt process.
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 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 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 central simple algebra over a finite field must be a matrix ring over that field.
* Every finite-dimensional reflexive algebra is hyper-reflexive.
Every linear function on a finite-dimensional space is continuous.

Every and simple
Every finite simple group is isomorphic to one of the following groups:
# 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.
Every simple module is indecomposable, but the converse is in general not true.
Every simple module is cyclic, that is it is generated by one element.
" 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 maximal outerplanar graph is the visibility graph of a simple polygon.
Every closed curve c on X is homologous to for some simple closed curves c < sub > i </ sub >, that is,
* Every automorphism of a central simple algebra is an inner automorphism ( follows from Skolem – Noether theorem ).
* 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.
Every simple ring R with unity is both left and right primitive.
# 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.
* Rule 2 Each column has a simple generic model: Every column can have its own meta-model

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 division ring is therefore a division algebra over its center.
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 finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian.
Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
Every state on a C *- algebra is of the above type.
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.
* Every Boolean algebra is a Heyting algebra, with given by.

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