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Some Related Sentences
Every and prime
Every positive integer n
> 1 can be represented
in exactly one way as
a product
of prime powers
:

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 prime ideal of A
is principal
.
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 non-zero characteristic
is a prime number
.
: Every Boolean algebra contains
a prime ideal.

*
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
ring other than
the trivial
ring contains
a maximal 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 principal
ideal domain is a unique factorization
domain ( UFD ).

#
Every principal
ideal domain is Noetherian
.

#
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
.

* Krull's theorem ( 1929 ):
Every ring with
a multiplicative identity has
a maximal ideal.
Every finitely generated
ideal of a Boolean ring is principal ( indeed
, ( x
, y )=( x + y + xy )).

*
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 >.

( DD1 )
Every nonzero proper
ideal factors into primes
.

( DD3 )
Every fractional
ideal of is invertible
.
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
.
Every element
of the Baer radical
is nilpotent
, so it is a nil
ideal.

In Leedskalnin's own publication A Book
in Every Home he implies his " Sweet Sixteen " was more
an ideal than
a reality
.

*
Every localization
of R at
a maximal ideal is a field
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 adiabat asymptotically approaches both
the V axis
and the P axis ( just like isotherms ).

*
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
.
Every polynomial
P in x corresponds
to a function
, ƒ ( x )

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
.

* Universal affirmative
: Every S
is a P.

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
.
Every order-preserving self-map f
of a cpo (
P, ⊥) has
a least fixpoint
.

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
.
Every other day
, starting on
the third day
, the player can go after
an A
. P. B
.

" 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.

There's One Born
Every Minute ( Los Angeles
, Ca
, U
. S
. A .: Jeremy
P. Tarcher
, Inc
, 1976
.
Every major Italian town or city has
a main
P. d
. S
.
0.561 seconds.