Page "Noetherian ring" ¶ 64
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## Some Related Sentences

Every and finitely-generated
* Every finitely-generated locally cyclic group is cyclic.
Every finitely-generated R-module is a direct sum of these.
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 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 localization of a commutative Noetherian ring is Noetherian.
* Every commutative semisimple ring must be a finite direct product of fields.
* 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 algebra
Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions.
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 real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions.
* 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 division ring is therefore a division algebra over its center.
Every continuous map f: X → Y induces an algebra homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ in C ( Y ).
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 associative algebra is obviously power-associative, but so are all other alternative algebras ( like the octonions, which are non-associative ) and even some non-alternative algebras like the sedenions.
Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.
Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
Every state on a C *- algebra is of the above type.
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 Heyting algebra with exactly one coatom is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new top.
* Every Boolean algebra is a Heyting algebra, with given by.

Every and over
The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.
Every six months the presidency rotates between the states, in an order predefined by the Council's members, allowing each state to preside over the body.
Every year many teams from all over Asia converge in Tokyo to compete.
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 field is, of course, one-dimensional over its center.
* Every June, Ellensburg hosts Dachshunds on Parade, an event that draws Dachshund dog owners from all over the Northwest.
Every year sidecar racers travel from all over the UK to Knockhill in Scotland to race in the prestigious ' Jock Taylor Memorial Trophy ' race.
Every day, a blessing is recited over the Lulav and the Etrog.
Every major commercial transport and almost every military jet built in the United States over the last 40 years has been tested in this facility.
Every car uses either a 5. 0 L Ford " Boss 302 " SVO or a 5. 0 L Chevrolet small block race-engine ( depending on the make )-capable of producing between 460 and 485 kW ( 620 — 650 bhp ) of power, but generally quoted as a little over 450 kW ( 600 bhp ) in race trim.
Every day all over the country, police officers are testifying.
Every character in the novel — both men and women — knows their place, and the traditional stereotypes of gender roles are repeated over and over again.
Every Turing machine computes a certain fixed partial computable function from the input strings over its alphabet.
* Every ninth year, people from all over Sweden assembled at the Temple at Uppsala.
Every year, millions of pilgrims from all over the world come to Częstochowa to see it.
Every Calan Mai, the two would fight over Creiddylad, until a battle on Judgement Day, in which the victor would keep her forever.
Controversy ensued over the use of the phrase " Every seat lost to the government is a seat sold to the Boers " as the Unionists waged a personalised campaign against Liberal critics of the war – some posters even portrayed Liberal MPs praising President Kruger and helping him to haul down the Union Jack.
Every year, the post office also creates a special Christmas hand-cancellation pictorial postmark for use during December, which also attracts mail from all over the world.
A quotation attributed to Kolmogorov is into English: " Every mathematician believes that he is ahead over all others.
Every August the Mountain Mama Road Bike Challenge brings cyclists from all over the country to challenge themselves on the steep roads of Highland County.
The argument over the quality of Fitzgerald ’ s translation of the Rubaiyat has, according to Dougan, diverted attention from a fuller understanding of the deeply esoteric message contained in Omar ’ s actual material – " Every line of the Rubaiyat has more meaning than almost anything you could read in Sufi literature ".

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