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Every Boolean ring R satisfies x ⊕ x

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## Some Related Sentences

Every and Boolean

__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.

__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__

__Boolean__algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for

__Boolean__algebras.

__Every__

__Boolean__algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.

__Every__complemented distributive lattice has a unique orthocomplementation and is in fact a

__Boolean__algebra.

__Every__finite

__Boolean__algebra can be represented as a whole power set-the power set of its set of atoms ; each element of the

__Boolean__algebra corresponds to the set of atoms below it ( the join of which is the element ).

__Every__( normal )

__Boolean__algebra with operators can be represented as a field of sets on a relational structure in the sense that it is isomorphic to the complex algebra corresponding to the field.

__Every__

__Boolean__algebra A has an essentially unique completion, which is a complete

__Boolean__algebra containing A such that every element is the supremum of some subset of A.

*

__Every__subset of a complete__Boolean__algebra has a supremum, by definition ; it follows that every subset also has an infimum ( greatest lower bound ).

Every and ring

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__topological

__ring__is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.

__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 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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