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* 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.

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## Some Related Sentences

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__

__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 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,

*****

__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**

**.**

# 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__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__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

**.**

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