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* 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.

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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**

**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 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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