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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.

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