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* In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i. e. M is contained in exactly 2 ideals of R, namely M itself and the entire ring R. Every maximal ideal is in fact prime.

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## Some Related Sentences

any and ring

One means to help

**the**birds occurs to me: Let**the**chimes**that**__ring__over Washington Square twice daily**,**discontinue__any__piece**of**music but one**.**
The morning hawk

**,**hungry for__any__eatable**,**killable**,**digestible item**,**kept his eyes on**the**__ring__**of**anchored ships**that**lay off**the**shores**in****the**bay**,**sheltered by**the**Jersey inlets**.**
`` The reason you are

**in****the**__ring__today**is**to show your ability to present to__any__judge**the**most attractive picture**of**your dog**that****the**skillful use**of**your aids can produce**.**
Bragi generously offers his sword

**,**horse**,****and****an**arm__ring__as peace gift but Loki only responds by accusing Bragi**of**cowardice**,****of**being**the**most afraid to fight**of**__any__**of****the**Æsir**and**Elves within**the**hall**.**
If

**a**fighter**is**knocked down during**the**fight**,**determined by whether**the**boxer touches**the**canvas floor**of****the**__ring__with__any__part**of**their body other than**the**feet as**a**result**of****the**opponent's punch**and**not**a**slip**,**as determined by**the**referee**,****the**referee begins counting until**the**fighter returns to his or her feet**and**can continue**.**
The definition works without

__any__changes if instead**of**vector spaces over**a**field F**,**we use modules over**a**commutative__ring__**R****.**It also can be easily generalized to n-ary functions**,**where**the****proper**term**is**multilinear**.**
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
As noted

**in****the**introduction**,**Bézout's identity works not only**in****the**__ring__**of**integers**,**but also**in**__any__other principal**ideal**domain ( PID ).*****The spectrum

**of**

__any__commutative

__ring__with

**the**Zariski topology (

**that**

**is**

**,**

**the**

**set**

**of**

**all**

**prime**

**ideals**)

**is**compact

**,**but never Hausdorff ( except

**in**trivial cases ).

Mail armour provided

**an**effective defence against slashing blows by**an**edged weapon**and**penetration by thrusting**and**piercing weapons ;**in****fact****a**study conducted at**the**Royal Armouries at Leeds concluded**that**" it**is**almost impossible to penetrate using__any__conventional medieval weapon " Generally speaking**,**mail's resistance to weapons**is**determined by four factors: linkage type ( riveted**,**butted**,**or welded ), material used ( iron versus bronze or steel ), weave density (**a**tighter weave needs**a**thinner weapon to surpass ),**and**__ring__thickness ( generally ranging from 18 to 14 gauge**in**most examples ).*****The reinforce: This portion

**of**

**the**piece

**is**frequently divided into

**a**first reinforce

**and**

**a**second reinforce

**,**but

**in**

__any__case

**is**marked as separate from

**the**chase by

**the**presence

**of**

**a**narrow circular reinforce

__ring__or band at its foremost end

**.**

Although most often used for matrices whose entries are real or complex numbers

**,****the**definition**of****the**determinant only involves addition**,**subtraction**and**multiplication**,****and**so it can be defined for square matrices with entries taken from__any__commutative__ring__**.**
This generalized Euclidean algorithm can be put to many

**of****the**same uses as Euclid's original algorithm**in****the**__ring__**of**integers:**in**__any__Euclidean domain**,**one can apply**the**Euclidean algorithm to compute**the**greatest common divisor**of**__any__two elements**.**
Especially

**,****the****fact****that****the**integers**and**__any__polynomial__ring__**in**one variable over**a**field are Euclidean domains such**that****the**Euclidean division**is**easily computable**is****of**basic importance**in**computer algebra**.**
Some authors also require

**the**domain**of****the**Euclidean function be**the****entire**__ring__**R**; this can always be accommodated by adding 1 to**the**values at**all**nonzero elements**,****and**defining**the**function to be 0 at**the**zero element**of****R****,**but**the**result**is**somewhat awkward**in****the**case**of**K**.**The definition**is**sometimes generalized by allowing**the**Euclidean function to take its values**in**__any__well-ordered**set**; this weakening does not affect**the**most important implications**of****the**Euclidean property**.**

any and R

For United States expenditures under subsections ( A )

**,**( B )**,**( D )**,**( E )**,**( F )**,**( H ) through (__R__)**of**Section 104**of****the**Act or under__any__**of**such subsections**,****the**rupee equivalent**of**$200 million**.**
Conversely

**,****a**subset__R__defines**a**binary function if**and**only if**,**for__any__x**in**X**and**y**in**Y**,**there exists**a**unique z**in**Z such**that**( x**,**y**,**z ) belongs to__R__**.**
So

**,**for example**,**while__R__< sup > n </ sup >**is****a**Banach space with respect to__any__norm defined on it**,**it**is**only**a**Hilbert space with respect to**the**Euclidean norm**.****In**simpler term

**,**Biotechnology

**is**

**the**research

**and**development

**in**

**the**laboratory

**that**involves bioinformatics for exploration

**,**extraction

**,**exploitation

**and**production from

__any__living organisms

**and**

__any__source

**of**biomass by means

**of**biochemical engineering where high value-added products could be planned ( reproduced by Biosynthesis

**,**for example ), fore-casted

**,**formulated

**,**developed

**,**manufactured

**and**marketed for

**the**purpose

**of**sustainable operations ( for

**the**return from bottomless initial investment on

__R__& D )

**and**gaining durable patents rights ( for exclusives rights for sales

**,**

**and**prior to this to receive national

**and**international approval from

**the**results on animal experiment

**and**human experiment

**,**especially on

**the**pharmaceutical branch

**of**biotechnology to prevent

__any__undetected side-effects on safety concerns by using

**the**products ), for more about

**the**biotechnology industry

**,**see

**.**

From Comrade Semichastny's speech I learn

**that****the**government**,**' would not put__any__obstacles**in****the**way**of**my departure from**the**U**.**S**.**S**.**__R__.' For me this**is**impossible**.*******

**In**

**the**cocountable topology on

__R__( or

__any__uncountable

**set**for

**that**matter ), no infinite

**set**

**is**compact

**.**

The space Q < sub > p </ sub >

**of**p-adic numbers**is**complete for__any__**prime**number p**.**This space completes Q with**the**p-adic metric**in****the**same way**that**__R__completes Q with**the**usual metric**.**
This

**is****a**generalization**of****the**Heine – Borel theorem**,**which states**that**__any__closed**and**bounded subspace S**of**__R__< sup > n </ sup >**is**compact**and**therefore complete**.**
Likewise

**,****the**problem**of**computing**a**quantity on**a**manifold which**is**invariant under differentiable mappings**is**inherently global**,**since__any__local invariant will be trivial**in****the**sense**that**it**is**already exhibited**in****the**topology**of**__R__< sup > n </ sup >.
Democide

**is****a**term revived**and**redefined by**the**political scientist__R__**.**J**.**Rummel as "**the**murder**of**__any__person or people by**a**government**,**including genocide**,**politicide**,****and**mass murder**.**
Although siblinghood

**is**symmetric ( if A**is****a**sibling**of**B**,**then B**is****a**sibling**of**A )**and**transitive on__any__3 distinct people ( if A**is****a**sibling**of**B**and**C**is****a**sibling**of**B**,**then A**is****a**sibling**of**C**,**provided A**is**not C ( Note**that**"**is****a**sibling**of**"**is**NOT**a**transitive relation**,**since A__R__B**,****and**B__R__A implies A__R__A by transitivity )), it**is**not reflexive ( A cannot be**a**sibling**of**A ).

any and maximal

** Hausdorff

__maximal__principle:**In**__any__partially ordered**set****,**every totally ordered subset**is****contained****in****a**__maximal__totally ordered subset**.**
The elements

**2****and**1 + √(− 3 ) are two "__maximal__common divisors " (**i****.****e****.**__any__common divisor which**is****a**multiple**of****2****is**associated to**2****,****the**same holds for 1 + √(− 3 )), but they are not associated**,**so there**is**no greatest common divisor**of****a****and**b**.**
It states

**that****in**__any__partially ordered**set****,**every totally ordered subset**is****contained****in****a**__maximal__totally ordered subset**.**
The Hausdorff

__maximal__principle states**that****,****in**__any__partially ordered**set****,**every totally ordered subset**is****contained****in****a**__maximal__totally ordered subset**.**
Here

**a**__maximal__totally-ordered subset**is**one**that****,**if enlarged**in**__any__way**,**does not remain totally ordered**.**
The holographic principle was inspired by black hole thermodynamics

**,**which implies**that****the**__maximal__entropy**in**__any__region scales with**the**radius squared**,****and**not cubed as might be expected**.**
For

__any__information rate**R**< C**and**coding error ε > 0**,**for large enough N**,**there exists**a**code**of**length N**and**rate ≥**R****and****a**decoding algorithm**,**such**that****the**__maximal__probability**of**block error**is**≤ ε ;**that****is****,**it**is**always possible to transmit with arbitrarily small block error**.**
For example

**,****the**direct sum**of****the****R**< sub >**i**</ sub > form**an****ideal**not**contained****in**__any__such A**,**but**the**axiom**of**choice gives**that**it**is****contained****in**some__maximal__**ideal**which**is****a**fortiori**prime****.**
Since

**all**__maximal__paths have**the**same number**of**black nodes**,**by property 5**,**this shows**that**no path**is**more than twice as long as__any__other path**.****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

**.**

For general groups

**,**Cauchy's theorem guarantees**the**existence**of****an**element**,****and**hence**of****a**cyclic subgroup**,****of**order__any__**prime**dividing**the**group order ; Sylow's theorem extends this to**the**existence**of****a**subgroup**of**order equal to**the**__maximal__power**of**__any__**prime**dividing**the**group order**.**
If r

**is****the**degree**of****the**primitive generator polynomial**,**then**the**__maximal__total block length**is****,****and****the**associated code**is**able to detect__any__single-bit or double-bit errors**.**
For

__any__normal modal logic L**,****a**Kripke model ( called**the**canonical model ) can be constructed**,**which validates precisely**the**theorems**of**L**,**by**an**adaptation**of****the**standard technique**of**using__maximal__consistent sets as models**.**
Given

**a****ring****R****and****a****proper****ideal**I**of****R**(**that****is**I ≠**R**), I**is****a**__maximal__**ideal****of****R**if__any__**of****the**following equivalent conditions hold:
The converse

**is**not always true: for example**,****in**__any__nonfield integral domain**the**zero**ideal****is****a****prime****ideal**which**is**not__maximal__**.**
For

**an****R**module A**,****a**__maximal__submodule**M****of**A**is****a**submodule**M**≠ A for which for__any__other submodule N**,**if**M**⊆ N ⊆ A then N =**M**or N = A**.**
A subset T

**is**totally ordered if for__any__s**,**t**in**T we have s ≤ t or t ≤ s**.**Such**a****set**T has**an**upper bound u**in**P if t ≤ u for**all**t**in**T**.**Note**that**u**is****an**element**of**P but need not be**an**element**of**T**.**An element m**of**P**is**called**a**__maximal__element ( or non-dominated ) if there**is**no element x**in**P for which m < x**.**
As

**a**rule**of**thumb**,**__any__sort**of**construction**that**takes as input**a**fairly general object ( often**of****an**algebraic**,**or topological-algebraic nature )**and**outputs**a**compact space**is**likely to use Tychonoff:**e****.**g.,**the**Gelfand space**of**__maximal__**ideals****of****a**commutative C*****algebra**,****the**Stone space**of**__maximal__**ideals****of****a**Boolean algebra**,****and****the**Berkovich spectrum**of****a**commutative Banach**ring****.**0.154 seconds.