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Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

from
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## Some Related Sentences

Every and prime

Hilbert's example

**:**"**the**assertion that either there**are**only finitely many__prime__numbers or there**are**infinitely many " ( quoted**in**Davis 2000**:**97 );**and**Brouwer's**:**"__Every__mathematical species**is**either finite or infinite**.**__Every__completely multiplicative function

**is**

**a**homomorphism

**of**monoids

**and**

**is**completely determined by its restriction

**to**

**the**

__prime__numbers

**.**

* 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__**.**__Every__twin

__prime__pair except ( 3

**,**5 )

**is**

**of**

**the**form ( 6n − 1

**,**6n + 1 ) for some natural number n

**,**

**and**with

**the**exception

**of**

**<**var

**>**n

**</**var

**>**= 1

**,**

**<**var

**>**n

**</**var

**>**must end

**in**0

**,**

**2**

**,**3

**,**5

**,**7

**,**or 8

**.**

__Every__third odd number

**is**divisible by 3

**,**

**which**requires that no three successive odd numbers can be

__prime__unless one

**of**them

**is**3

**.**

* Hardy

**and**Littlewood listed as their Conjecture I**:**"__Every__large odd number ( n**>**5 )**is****the**sum**of****a**__prime__**and****the**double**of****a**__prime__**.**__Every__strong probable

__prime__

**to**base

**a**

**is**

**also**

**an**Euler probable

__prime__

**to**

**the**same base

**,**but not vice versa

**.**

__Every__closed 3-manifold has

**a**

__prime__decomposition

**:**this means

**it**

**is**

**the**connected sum

**of**

__prime__three-manifolds ( this decomposition

**is**essentially unique except for

**a**small problem

**in**

**the**case

**of**non-orientable manifolds ).

**:**

__Every__oriented

__prime__closed 3-manifold can be cut along tori

**,**

**so**that

**the**interior

**of**each

**of**

**the**resulting manifolds has

**a**geometric structure with finite volume

**.**

*

__Every__number**of****the**form**2****<**sup**>**m**</**sup**>**p for**a**natural number m**and****a**__prime__number p such that p**<****2****<**sup**>**m + 1**</**sup**>****is****also**semiperfect**.**__Every__group

**of**

__prime__order

**is**cyclic

**,**since Lagrange's theorem implies that

**the**cyclic subgroup generated by

Iwasawa worked with so-called-extensions

**:**infinite extensions**of****a**number**field**with Galois group**isomorphic****to****the**additive group**of**p-adic integers for some__prime__p**.**__Every__closed subgroup**of****is****of****the**form**,****so**by Galois theory**,**a-extension**is****the**same thing as**a**tower**of**fields such that**.**__Every__imperfect

**field**

**is**necessarily transcendental over its

__prime__subfield (

**the**minimal subfield ), because

**the**latter

**is**perfect

**.**

A finitely-generated abelian group

**is**indecomposable if**and**only if**it****is****isomorphic****to**Z or**to****a**factor group**of****the**form for some__prime__number p**and**some positive integer n**.**__Every__finitely-generated abelian group**is****a**direct sum**of**( finitely many ) indecomposable abelian groups**.**
*

__Every__non-split**,**__prime__**,**alternating link that**is**not**a**torus link**is**hyperbolic by**a**result**of**William Menasco**.**

Every and ideal

*

__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__character

**is**automatically continuous from A

**to**C

**,**since

**the**kernel

**of**

**a**character

**is**

**a**

**maximal**

__ideal__

**,**

**which**

**is**closed

**.**

#

__Every__finitely generated__ideal__**of**A**is**principal ( i**.**e., A**is****a**Bézout**domain**)**and**A satisfies**the**ascending chain condition on principal**ideals****.**__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

**.**

*

__Every__left__ideal__I**in****R****is**finitely generated**,**i**.**e**.**there exist elements**a****<****sub****>**1**</****sub****>,**...,**a****<****sub****>**n**</****sub****>****in**I such that I = Ra**<****sub****>**1**</****sub****>**+ ... + Ra**<****sub****>**n**</****sub**>.__Every__

**field**

**of**either type can be realized as

**the**

**field**

**of**fractions

**of**

**a**Dedekind

**domain**

**in**

**which**every non-zero

__ideal__

**is**

**of**finite index

**.**

In Leedskalnin's own publication A Book

**in**__Every__Home he implies his " Sweet Sixteen " was more**an**__ideal__than**a**reality**.**__Every__meal can be eaten without any cooking ; when circumstances permit

**,**

**the**

__ideal__method

**of**preparation

**is**

**to**cook

**the**entrees

**in**

**a**pressure cooker

**,**heated on

**the**standard issue Coleman stove

**,**or by simply boiling

**the**rations

**in**its package

**in**water

**.**

__Every__free spirit looked upon her as protectoress

**and**

__ideal__.... Marguerite was

**the**embodiment

**of**charity

**.**

Every and P

*

__Every__quadratic Bézier curve**is****also****a**cubic Bézier curve**,****and**more generally**,**every degree n Bézier curve**is****also****a**degree m curve for any m**>**n**.**In detail**,****a**degree n curve with control points__P__**<****sub****>**0**</****sub****>,**…,__P__**<****sub****>**n**</****sub****>****is**equivalent ( including**the**parametrization )**to****the**degree n + 1 curve with control points__P__'<**sub****>**0**</****sub****>,**…,__P__'<**sub****>**n + 1**</****sub****>,**where**.**
The definition

**of**permanent agriculture as that**which**can be sustained indefinitely was supported by Australian__P__**.**A**.**Yeomans**in**his 1973 book Water for__Every__Farm**.**
Jeffrey

__P__**.**Dennis**,**author**of****the**journal article " The Same Thing We Do__Every__Night**:**Signifying Same-Sex Desire**in**Television Cartoons ," argued that SpongeBob**and**Sandy**are**not romantically**in**love**,**while adding that he believed that SpongeBob**and**Patrick "**are**paired with arguably erotic intensity**.**__Every__object would

**also**have

**a**read timestamp

**,**

**and**if

**a**transaction T

**<**

**sub**

**>**i

**</**

**sub**

**>**wanted

**to**write

**to**object

__P__

**,**

**and**

**the**timestamp

**of**that transaction

**is**earlier than

**the**object's read timestamp ( TS ( T

**<**

**sub**

**>**i

**</**

**sub**>)

**<**RTS (

__P__)),

**the**transaction T

**<**

**sub**

**>**i

**</**

**sub**

**>**

**is**aborted

**and**restarted

**.**

__Every__such line meets

**the**sphere

**of**radius one centered

**in**

**the**origin exactly twice

**,**say

**in**

__P__= ( x

**,**y

**,**z )

**and**its antipodal point (− x

**,**− y

**,**− z ).

*

__Every__irreducible closed subset**of**__P__**<**sup**>**n**</**sup >( k )**of**codimension one**is****a**hypersurface ; i**.**e.,**the**zero set**of**some homogeneous polynomial**.**
In mathematics

**,****an**integer-valued polynomial (**also**known as**a**numerical polynomial )__P__( t )**is****a**polynomial whose value__P__( n )**is****an**integer for every integer n**.**__Every__polynomial with integer coefficients**is**integer-valued**,**but**the**converse**is**not true**.**
" Jeffrey

__P__**.**Dennis**,**author**of**" The Same Thing We Do__Every__Night**:**Signifying Same-Sex Desire**in**Television Cartoons ," argued that**the**romantic connection between Velma**and**Daphne Blake**is**" mostly wishful thinking " because Velma**and**Daphne " barely acknowledge each other's existence**.**
#

__Every__submodule**of**M**is****a**direct summand**:**for every submodule N**of**M**,**there**is****a**complement__P__such that M = N ⊕__P__**.**0.605 seconds.