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Every finitely generated ideal of a Boolean ring is principal ( indeed, ( x, y )=( x + y + xy )).

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## Some Related Sentences

Every and finitely

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____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____finitely__**generated**group with**a**recursively enumerable presentation and insoluble word problem**is****a**subgroup**of****a**__finitely__presented group with insoluble word problem__Every__subset

**of**

**a**nowhere dense set

**is**nowhere dense

**,**and the union

**of**

__finitely__many nowhere dense sets

**is**nowhere dense.

*

__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__

__finitely__presented group

**is**recursively presented

**,**but there are recursively presented groups that cannot be

__finitely__presented.

__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 >.

" I can now state the physical version

**of**the Church-Turing principle: '__Every____finitely__realizable physical system can be perfectly simulated by**a**universal model computing machine operating by finite means.
*

__Every__substructure**is**the union**of**its__finitely__**generated**substructures ; hence Sub**(**A )**is**an algebraic lattice.
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.
This

**is**up to isomorphism the only indecomposable module over R.__Every__left R-module**is****a**direct sum**of****(**__finitely__or infinitely many ) copies**of**this module K < sup > n </ sup >.__Every__

**(**bounded ) convex polytope

**is**the image

**of**

**a**simplex

**,**as every point

**is**

**a**convex combination

**of**the

**(**

__finitely__many ) vertices.

Every and generated

__Every__group

**of**prime order

**is**cyclic

**,**since Lagrange's theorem implies that the cyclic subgroup

__generated__by

__Every__lattice in can be

__generated__from

**a**basis for the vector space by forming all linear combinations with integer coefficients.

__Every__film today

**,**whether it be live-action

**,**computer

__generated__

**,**or traditional hand-drawn animation

**is**made up

**of**hundreds

**of**individual shots that are all placed together during editing to form the single film that

**is**viewed by the audience.

A large part

**of**the interest__generated__was due to the eclectic mix**of**samples and influences**,**evident on__Every__Man and Woman**,**which was dedicated to Dewey Bunnell**of**the band America.__Every__ticket

__generated__by the system has persistence or " history " showing what happened to the ticket within its life cycle.

__Every__profile also captures some non-crime statistics: the amount

**of**overtime

__generated__by members

**of**the command

**,**the number

**of**department vehicle accidents

**,**absence rates due to sick time and line-of-duty injuries

**,**and the number

**of**civilian complaints lodged against members

**of**the unit.

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.

* 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__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__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.

__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.

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