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* Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication.

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## Some Related Sentences

Any and ring

__Any__legal wrestler is open to attack from any direction at any time, including when they are downed, as long as they are within the

__ring__area enclosed by the

__ring__ropes

**.**

__Any__number system that

**forms**

**a**

**commutative**

__ring__— for instance, the integers, the real numbers,

**and**the complex numbers — can be extended to

**a**wheel

**in**which division by zero is always possible ; however,

**in**such

**a**case, " division " has

**a**slightly different meaning

**.**

*****

__Any__

__ring__

**R**can be considered as

**a**one-object preadditive category ; the category

**of**left modules over

**R**is the same as the additive functor category Add (

**R**, Ab ) ( where Ab denotes the category

**of**abelian groups ),

**and**the category

**of**right R-modules is Add (

**R**< sup > op </ sup >, Ab ).

__Any__prime number p gives rise to

**an**ideal pO < sub > K </ sub >

**in**the

__ring__

**of**integers O < sub > K </ sub >

**of**

**a**quadratic field K

**.**

*****

__Any__algebra over

**a**

**commutative**

__ring__K may be regarded as

**a**purely even superalgebra over K ; that is, by taking A < sub > 1 </ sub > to be trivial

**.**

__Any__n-dimensional formal group law gives

**an**n dimensional Lie algebra over the

__ring__

**R**, defined

**in**terms

**of**the quadratic part F < sub > 2 </ sub >

**of**the formal group law

**.**

Any and matrices

*****

__Any__simply connected solvable Lie group is isomorphic to

**a**closed subgroup

**of**the group

**of**invertible upper triangular

__matrices__

**of**some rank,

**and**any finite dimensional irreducible representation

**of**such

**a**group is 1 dimensional

**.**

*****

__Any__simply connected nilpotent Lie group is isomorphic to

**a**closed subgroup

**of**the group

**of**invertible upper triangular

__matrices__

**with**1's on the diagonal

**of**some rank,

**and**any finite dimensional irreducible representation

**of**such

**a**group is 1 dimensional

**.**

__Any__element

**in**the direct sum

**of**two vector spaces

**of**

__matrices__can be represented as

**a**direct sum

**of**two

__matrices__

**.**

__Any__unipotent algebraic group is isomorphic to

**a**closed subgroup

**of**the group

**of**upper triangular

__matrices__

**with**diagonal entries 1,

**and**conversely any such subgroup is unipotent

**.**

__Any__2 × 2

**matrix**can be written

**in**the form, where I is the identity

**matrix**, σ < sub > i </ sub > are the Pauli

__matrices__

**and**u is

**a**unit vector

**.**

Any and with

__Any__posse riding down the street to demand Blue Throat's surrender would be wiped out

__with__one deadly burst

**of**fire

**.**

__Any__abilities I may have were achieved

**in**their present shape from experience

**in**sharing

**in**the growth

**and**control

**of**my business, coupled

__with__raising my family

**.**

__Any__attempt to reconcile this statement

**of**the central issue

**in**the campaign

**of**1956

__with__the nature

**of**the man who could not conceive it as the central issue will at least resolve our confusions about the chaotic

**and**misleading results

**of**the earnestness

**of**both doctors

**and**President

**in**

**a**situation which should never have arisen

**.**

__Any__claimant whose claim is denied, or is approved for less than the full amount

**of**such claim, shall be entitled,

**under**such regulations as the Commission may prescribe, to

**a**hearing before the Commission, or its duly authorized representatives,

__with__respect to such claim

**.**

__Any__irregularity on the piston heads will make it impossible,

__with__normal means, to determine the final combustion chamber volume because the volume displaced by the piston heads cannot be readily computed

**.**

__Any__reputable French interne can supply you

__with__

**a**dozen similar instances,

**and**I'll presently recount

**a**case out

**of**my own personal experience, but, for the moment, let's resume our catalogue

**.**

__Any__

**of**the antithyroid drugs,

**of**either type, if given

**in**large enough doses for

**a**long period

**of**time will cause goitre, owing to inhibition

**of**thyroid hormone synthesis,

__with__production

**of**hypothyroidism

**.**

__Any__adjustments which are made, Mr. Grenier said earlier this month, will appear on the balance

**of**the tax bill since most

**of**the town's taxpayers take the option

**of**paying quarterly

__with__the balance due next year

**.**

__Any__shareholder

**of**any

**of**these funds who finds

**a**mistake

**in**his account certainly should get

**in**touch

__with__the bank

**.**

__Any__musician playing Beethoven here, where Beethoven was born, is likely to examine his own interpretations

__with__special care

**.**

Most

**of**the C code being written nowadays is based on ANSI C**.**__Any__program written only**in**standard C**and**without any hardware dependent assumptions is virtually guaranteed to compile correctly on any platform__with__**a**conforming C implementation**.**__Any__company incorporating Bluetooth wireless technology into products, using the technology to offer goods

**and**services or simply re-branding

**a**product

__with__Bluetooth technology must become

**a**member

**of**the Bluetooth SIG

**.**

__Any__codex

**of**Bede's Easter cycle is normally found together

__with__

**a**codex

**of**his " De Temporum Ratione ".

Love

**and**Erlandson continued to pursue__with__the band,**and**released the single " Be A Man "—**an**outtake from the Celebrity Skin sessions — for the soundtrack**of**the Oliver Stone film__Any__Given Sunday ( 1999 ).__Any__differential area

__with__normal vector

**of**

**a**given internal surface area, bounding

**a**portion

**of**the body, experiences

**a**contact force arising from the contact between both portions

**of**the body on each side

**of**,

**and**it is given by

#

__Any__collection**of**closed subsets**of**X__with__the finite intersection property has nonempty intersection**.**__Any__season ticket holder could put themselves forward for election,

__with__

**a**certain number

**of**nominations,

**and**votes were cast by all season ticket holders over the age

**of**18

**.**

__Any__pellets which are too small to be finished granules

**of**couscous

**and**fall through the sieve will be again rolled

**and**sprinkled

__with__dry semolina

**and**rolled into pellets

**.**

__Any__successes that the Nationalists did make, however, were met

__with__constant political

**and**military upheavals

**.**

__Any__piece

**of**DNA

__with__the point centromere DNA sequence on it will typically form

**a**centomere if present

**in**the appropriate species

**.**

#

__Any__experience which is inconsistent__with__the organization**of**the structure**of**the self may be perceived as**a**threat,**and**the more**of**these perceptions there are, the more rigidly the self structure is organized to maintain itself**.**__Any__rational number

__with__

**a**denominator whose only prime factors are 2

**and**/ or 5 may be precisely expressed as

**a**decimal fraction

**and**has

**a**finite decimal expansion

**.**

__Any__units

**of**angular measure can be used for declination, but it is customarily measured

**in**the degrees ( ° ), minutes ( ' ),

**and**seconds ( " )

**of**sexagesimal measure,

__with__90 ° equivalent to circle

**.**

__Any__ellipse can be obtained by rotation

**and**translation

**of**

**a**canonical ellipse

__with__the proper semi-diameters

**.**

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