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* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.

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## Some Related Sentences

Every and polynomial

__Every__output of an encoder can be described by its own transfer function, which

**is**closely related to the generator

__polynomial__

**.**

*****

__Every__root of

**a**monic

__polynomial__whose coefficients are algebraic integers

**is**itself an algebraic integer

**.**

__Every__

__polynomial__in can be factorized into polynomials that are irreducible over F

**.**This factorization

**is**unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the

**ring**of polynomials over

**a**field

**is**

**a**unique factorization domain whose units are the nonzero constant polynomials ).

__Every__delta operator ' has

**a**unique sequence of " basic polynomials ",

**a**

__polynomial__sequence defined by three conditions:

*****

__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__Jacobi-like

__polynomial__sequence can have its domain shifted and / or scaled so that its interval of orthogonality

**is**, and has Q

*****

__Every__Laguerre-like

__polynomial__sequence can have its domain shifted, scaled, and / or reflected so that its interval of orthogonality

**is**, and has Q =

*****

__Every__Hermite-like

__polynomial__sequence can have its domain shifted and / or scaled so that its interval of orthogonality

**is**, and has Q

__Every__field and every

__polynomial__

**ring**over

**a**field ( in arbitrarily many variables )

**is**

**a**reduced

**ring**

**.**

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 and ring

__Every__Boolean algebra ( A, ∧, ∨) gives rise to

**a**

__ring__( A, +, ·) by defining

**a**+ b := (

**a**∧ ¬ b ) ∨ ( b ∧ ¬

**a**) = (

**a**∨ b ) ∧ ¬(

**a**∧ b ) ( this operation

**is**called symmetric difference in the case of sets and XOR in the case of logic ) and

**a**· b :=

**a**∧ b

**.**The zero element of this

__ring__coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the

__ring__

**is**the 1 of the Boolean algebra

**.**

In Norse mythology, Draupnir ( Old Norse " the dripper ")

**is****a**gold__ring__possessed by the god Odin with the ability to multiply itself:__Every__ninth night eight new rings ' drip ' from Draupnir, each one of the same size and weight as the original**.**__Every__module over

**a**division

__ring__has

**a**basis ; linear maps between finite-dimensional modules over

**a**division

__ring__can be described by matrices, and the Gaussian elimination algorithm remains applicable

**.**

__Every__objective has

**a**different size

__ring__, so for every objective another condenser setting has to be chosen

**.**

__Every__match must be assigned

**a**rule keeper known as

**a**referee, who

**is**the final arbitrator ( In multi-man lucha libre matches, two referees are used, one inside the

__ring__and one outside ).

*****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 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__topological

__ring__

**is**

**a**topological group ( with respect to addition ) and hence

**a**uniform space in

**a**natural manner

**.**

Every and R

__Every__holomorphic function can be separated into its real and imaginary parts, and each of these

**is**

**a**solution of Laplace's equation on

__R__

**<**sup

**>**2

**</**sup >.

__Every__vector v in determines

**a**linear map from

__R__to taking 1 to v, which can be thought of as

**a**Lie algebra homomorphism

**.**

__Every__binary relation

__R__on

**a**set S can be extended to

**a**preorder on S by taking the transitive closure and reflexive closure,

__R__

**<**sup >+=</ sup >.

*****

__Every__separable metric space

**is**isometric to

**a**subset of C (), the separable Banach space of continuous functions →

__R__, with the supremum norm

**.**

__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__random vector gives rise to

**a**probability measure on

__R__

**<**sup

**>**

**n**

**</**sup

**>**with the Borel algebra as the underlying sigma-algebra

**.**

*****

__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__non-empty set of left ideals of

__R__, partially ordered by inclusion, has

**a**maximal element with respect to set inclusion

**.**

__Every__year since 1982, the W

**.**C

**.**Handy Music Festival

**is**held in the Florence / Sheffield / Muscle Shoals area, featuring blues, jazz, country, gospel, rock music and

__R__& B

**.**

__Every__adult citizen of this small settlement signed the small petition ; E

**.**K Dyer and his wife, William Johnson, Joseph Otis and his wife, Hiram Walker and his wife, Joseph Pease and

__R__

**.**H

**.**Valentine

**.**

__Every__smooth submanifold of

__R__

**<**sup

**>**

**n**

**</**sup

**>**has an induced Riemannian metric g: the inner product on each tangent space

**is**the restriction of the inner product on

__R__

**<**sup

**>**

**n**

**</**sup >.

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