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For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies

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## Some Related Sentences

For and case

__For__

**the**

__case__

**of**Jupiter

**,**

**the**radio emission spectrum

**is**definitely not like

**the**spectrum

**of**

**a**black-body radiator

**,**

**and**it seems very likely

**that**

**the**radiation reaching

**the**earth

**is**

**a**combination

**of**thermal radiation from

**the**atmosphere

**and**non-thermal components.

__For__if

**such**were

**the**

__case__

**,**either

**the**plane

**of**

**the**two lines would meet **zg

**in**more than K points or

**,**alternatively

**,**

**the**order

**of**

**the**image regulus

**of**

**the**pencil determined by

**the**two lines would be

**too**high.

__For__

**the**

__case__

**of**

**a**purely inertial autonavigator consisting

**of**three restrained gyros

**,**

**a**coordinate system

**is**used

**where**

**the**sensitive axis

**of**

**the**X accelerometer

**is**parallel to

**the**east-west direction at

**the**

**base**point

**,**

**and**

**the**Y accelerometer sensitive axis

**is**parallel to

**the**north-south direction at

**the**

**base**point.

__For__

**an**ideal gas

**,**

**the**temperature remains constant because

**the**internal energy only depends on temperature

**in**

**that**

__case__.

__For__private suits only

**the**victims or their families could prosecute

**,**while

**for**public suits anyone ( ho boulomenos

**,**' whoever wants to ' i. e.

**any**citizen with full citizen rights ) could bring

**a**

__case__since

**the**issues

**in**these major suits were regarded as affecting

**the**community as

**a**whole.

__For__example

**a**two dimensional array with three rows

**and**four columns might provide access to

**the**element at

**the**2nd row

**and**4th column by

**the**expression: (

**in**

**a**row major language )

**and**(

**in**

**a**column major language )

**in**

**the**

__case__

**of**

**a**zero-based indexing system.

__For__

**the**frequent

__case__

**of**propositional logic

**,**

**the**problem

**is**decidable but Co-NP-complete

**,**

**and**hence only exponential-time algorithms are believed to exist

**for**general proof tasks.

__For__

**the**simple

__case__

**where**

**the**direction

**of**

**the**light beam along

**the**line SE

**is**perpendicular to

**the**motion

**of**

**the**observer then only light emitted at

**an**angle equal to

**the**arc cosine

**of**

**the**ratio

**of**

**the**observer's speed to

**the**speed

**of**light will reach

**the**telescope.

__For__

**the**

__case__

**of**

**an**object

**that**

**is**small compared with

**the**radial distance to its axis

**of**rotation

**,**

**such**as

**a**tin

**can**swinging from

**a**long string or

**a**planet orbiting

**in**

**a**circle around

**the**Sun

**,**

**the**angular momentum

**can**be expressed as its linear momentum

**,**

**,**crossed by its position from

**the**origin

**,**r. Thus

**,**

**the**angular momentum L

**of**

**a**particle with respect to some point

**of**origin

**is**

__For__larger quantities one must go to

**a**beverage distributor

**which**sells beer only by

**the**

__case__or keg.

__For__example

**,**bacteria catalyse

**the**breakdown

**of**

**the**mineral pyrite ( FeS

**<**

**sub**

**>**2

**</**

**sub**>) by oxidising

**the**sulfur

**and**metal (

**in**this

__case__ferrous iron

**,**( Fe

**<**sup

**>**2 +</ sup >)) using oxygen.

__For__

__case__

**of**single stranded DNA / RNA units

**of**nucleotides are used

**,**abbreviated nt ( or knt

**,**Mnt

**,**Gnt ), as they are not paired.

__For__example

**in**

**the**

__case__

**of**anthrax

**,**it

**is**likely

**that**by 24 – 36 hours after

**an**attack

**,**some small percentage

**of**individuals ( those with compromised immune system or who had received

**a**large dose

**of**

**the**organism due to proximity to

**the**release point ) will become ill with classical symptoms

**and**signs ( including

**a**virtually unique chest X-ray finding

**,**often recognized by public health officials if they receive timely reports ).

__For__example

**,**most areas

**of**law

**in**most Anglo-American jurisdictions include " statutory law " enacted by

**a**legislature

**,**" regulatory law " promulgated by executive branch agencies pursuant to delegation

**of**rule-making authority from

**the**legislature

**,**

**and**common law or "

__case__law ", i. e., decisions issued by courts ( or quasi-judicial tribunals within agencies ).

__For__example

**,**civil law

**can**be clearer than

__case__law when

**the**legislature has had

**the**foresight

**and**diligence to address

**the**precise set

**of**facts applicable to

**a**particular situation.

__For__example

**,**

**the**1972 Constitution designates

**the**Prime Minister as constitutional successor

**of**

**the**Head

**of**State

**in**

__case__

**of**incapacity

**,**death

**,**resignation or unaccountable absence

**of**

**the**incumbent.

__For__example

**,**

**in**one

__case__

**,**

**a**prominent Chicana feminist writer

**and**poet has indicated

**the**following subjective meaning through her creative work.

__For__example

**in**

**the**Netherlands

**,**

**the**law assumes

**the**stronger participant ( e. g.

**a**car driver )

**is**liable

**in**

**the**

__case__

**of**

**an**accident with

**a**weaker participant ( e. g.

**a**cyclist ) unless it

**can**be proved

**that**

**the**cyclist's behavior could not have been expected.

For and non-commutative

__For__example

**,**

**in**some groups

**,**

**the**

**group**operation

**is**commutative

**,**

**and**this

**can**be asserted with

**the**introduction

**of**

**an**additional axiom

**,**but without this axiom

**we**

**can**do quite well developing (

**the**more general )

**group**theory

**,**

**and**

**we**

**can**even take its negation as

**an**axiom

**for**

**the**study

**of**

__non-commutative__groups.

__For__square matrices with entries

**in**

**a**

__non-commutative__

**ring**

**,**

**for**instance

**the**quaternions

**,**there

**is**no unique definition

**for**

**the**determinant

**,**

**and**no definition

**that**has all

**the**usual properties

**of**determinants over commutative rings.

__For__

**the**element A ( t ) from

**the**algebra ( set g with

**the**

__non-commutative__product *),

**where**t

**is**

**the**" time parameter ",

**the**ordered exponential

__For__

**the**duality between σ-finite measure spaces

**and**commutative von Neumann algebras

**,**noncommutative von Neumann algebras are called

__non-commutative__measure spaces.

__For__

**a**

__non-commutative__

**ring**

**,**

**the**determinant cannot be defined

**,**but

**the**

**map**GL ( A ) → K

**<**

**sub**

**>**1

**</**

**sub**>( A ) generalizes

**the**determinant.

For and base

__For__

**a**brief period each year

**,**

**the**rays

**of**

**the**sun are warm enough to melt some

**of**

**the**snows piled

**a**mile deep at

**the**

__base__

**of**

**the**headwalls

**,**

**and**then

**the**pinnacles glisten

**in**

**the**daytime at high noon

**,**

**and**billions

**of**gallons

**of**water begin their slow seepage under

**the**glaciers

**and**across

**the**rockstrewn hanging valleys on their long

**,**meandering journey to

**the**sea -- running east past

**the**sky-carving massifs

**of**Gurla Mandhata

**and**Kemchenjunga

**,**then turning south

**and**curling down through

**the**jungles

**of**Assam

**,**past

**the**Khasi Hills

**,**

**and**into Bengal

**,**past Sirinjani

**and**Madaripur

**,**until

**the**hard water

**of**

**the**melting snows mingles with

**the**soft drainage

**of**fields

**and**at length fans out to meld with

**the**teeming salt depths

**of**

**the**Bay

**of**Bengal.

__For__

**the**States

**which**maintain two separate agencies -- one

**for**

**the**vocational rehabilitation

**of**

**the**blind

**,**

**and**one

**for**

**the**rehabilitation

**of**persons other than

**the**blind --

**the**Act specifies

**that**their minimum (

__base__) allotment shall be divided between

**the**two agencies

**in**

**the**same proportion as it was divided

**in**fiscal year 1954.

__For__

**a**vector with linear addressing

**,**

**the**element with index i

**is**located at

**the**address B + c · i

**,**

**where**B

**is**

**a**fixed

__base__address

**and**c

**a**fixed constant

**,**sometimes called

**the**address increment or stride.

__For__example

**,**if

**the**array has five elements

**,**indexed 1 through 5

**,**

**and**

**the**

__base__address B

**is**replaced by B − 30c

**,**then

**the**indices

**of**those same elements will be 31 to 35.

__For__

**a**time

**in**

**the**19th century

**,**stolen bases were credited when

**a**baserunner reached

**an**extra

__base__on

**a**

__base__hit from another player.

__For__Chrétien

**,**Arthur's chief court was

**in**Caerleon

**in**Wales ; this was

**the**king's primary

__base__

**in**Geoffrey

**of**Monmouth's Historia Regum Britanniae

**and**subsequent literature.

__For__instance

**,**

**the**largest human chromosome

**,**chromosome number 1

**,**

**is**approximately 220 million

__base__pairs long.

__For__these reasons

**,**it

**is**often considered

**a**party

**of**

**the**centre-left

**and**its

__base__consists largely

**of**

**the**working class.

__For__Husserl

**a**sentence has

**a**proposition or judgment as its meaning

**,**

**and**refers to

**a**state

**of**affairs

**which**has

**a**situation

**of**affairs as

**a**reference

__base__.

__For__elliptic-curve-based protocols

**,**it

**is**assumed

**that**finding

**the**discrete logarithm

**of**

**a**random elliptic curve element with respect to

**a**publicly known

__base__point

**is**infeasible.

__For__Golb

**,**

**the**amount

**of**documents

**is**

**too**extensive

**and**includes many different writing styles

**and**calligraphies ;

**the**ruins seem to have been

**a**fortress

**,**used as

**a**military

__base__

**for**

**a**very long period

**of**timeincluding

**the**1st centuryso they could not have been inhabited by

**the**Essenes ;

**and**

**the**large graveyard excavated

**in**1870

**,**just 50 metres east

**of**

**the**Qumran ruins was made

**of**over 1200 tombs

**that**included many women

**and**childrenPliny clearly wrote

**that**

**the**Essenes

**that**lived near

**the**Dead Sea " had not one woman

**,**had renounced all pleasure ...

**and**no one was born

**in**their race ".

In expert system technology

**,****the**knowledge__base__**is**expressed with natural language rules IF ... THEN ...__For__examples:__For__instance

**,**1 / 5 cannot be represented exactly as

**a**floating point number using

**a**binary

__base__but

**can**be represented exactly using

**a**decimal

__base__( 0. 2

**,**or 2x10 ^- 1 ).

__For__many years

**,**granite

**and**volcanic rocks have been quarried locally

**for**road

__base__metal

**,**riprap

**,**armour stone

**and**asphalt

**,**although

**the**main purpose now

**is**

**for**concrete aggregates.

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