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Page "Artin–Wedderburn theorem" ¶ 8
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## Some Related Sentences

Every and commutative
* Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra.
* Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
* Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional.
* Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian.
* Every localization of a commutative Noetherian ring is Noetherian.
* Every reduced commutative k-algebra A is a separable algebra ; i. e., is reduced for every field extension F / k.
Every commutative von Neumann algebra on a separable Hilbert space is isomorphic to L < sup >∞</ sup >( X ) for some standard measure space ( X, μ ) and conversely, for every standard measure space X, L < sup >∞</ sup >( X ) is a von Neumann algebra.
Every symmetric association scheme is commutative.

Every and semisimple
Every semisimple ring is von Neumann regular, and a left ( or right ) Noetherian von Neumann regular ring is semisimple.
Every ring which is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself.

Every and ring
** Every unital ring other than the trivial ring contains a maximal ideal.
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 division ring is therefore a division algebra over its center.
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.
* Krull's theorem ( 1929 ): Every ring with a multiplicative identity has a maximal ideal.
Every Boolean ring R satisfies x ⊕ x
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 finitely generated ideal of a Boolean ring is principal ( indeed, ( x, y )=( x + y + xy )).
Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.
Every prostaglandin contains 20 carbon atoms, including a 5-carbon ring.
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 must
Every member of the family must have a vital place in its life.
Every doll in the house must be going with her ''.
* Every collecting institution must develop an emergency plan to protect its collections and train staff to carry it out.
* Every institution must assign responsibility for caring for collections to members of its staff.
Every cleric in Holy Orders and every member of a religious order must publicly join in or privately read aloud ( i. e. using the lips as well as the eyes — it takes about two hours in this way ) the whole of the Breviary services allotted for each day.
* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.
Every team that attends must qualify and receive a bid at a partner company's competition.
Every DNS zone must be assigned a set of authoritative name servers that are installed in NS records in the parent zone, and should be installed ( to be authoritative records ) as self-referential NS records on the authoritative name servers.
Every sentence must contain formal words to designate what Husserl calls " formal categories ".
Every system must have a class designated as " root ", with one of its creation procedures designated as " root procedure ".
Every SE must be registered in the State where it has its registered office, in a register designated by the law of that State.
Every effort must be made to gather and organize useful intelligence.
In cybernetics, the Good Regulator or Conant-Ashby theorem is stated " Every Good Regulator of a system must be a model of that system ".
* Tested: Every change must be tested in a safe test environment, which closely reflects the actual production environment, before the change is applied to the production environment.
Every U. S. jurisdiction has its own regulations regarding what, precisely, must be said to a person arrested or placed in a custodial situation.
Every adult, healthy, sane Muslim who has the financial and physical capacity to travel to Mecca and can make arrangements for the care of his / her dependants during the trip, must perform the Hajj once in a lifetime.
* Every formal address or message " to the nation " or to either or both Houses of the Oireachtas must have prior approval of the Government.
Every company must deliver an annual return to Companies House at least once every twelve months.
Every karma produces a result which must be experienced either in this or some future life.
The moral law stands on an entirely different foundation from the ceremonial or ritual law .... Every part of this law must remain in force upon all mankind and in all ages.
Historian Barry Adam notes, " Every social movement must choose at some point what to retain and what to reject out of its past.
The last two lines of the Act IV-scene 2 funeral song may also have inspired the lines W. H. Auden, the librettist for Stravinsky's The Rake's Progress, puts into the mouth of Anne Truelove at the end of the opera: " Every wearied body must late or soon return to dust ".
Every time you move to a new page, however, the server must believe that you are the same person who originally signed in ( otherwise it will refuse ).
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.

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