 Page "Semisimple module" ¶ 23
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Every and ring ** Every unital ring other than the trivial ring contains a maximal ideal. * Every polynomial ring R ..., x < sub > n </ sub > is a commutative R-algebra. 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 finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. * Every localization of a commutative Noetherian ring is Noetherian. 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 which Every new scandal which would provide more `` copy '' for Marshall's pen would thus mean more publicity for Welch. Every dream, and this is true of a mental image of any type even though it may be readily interpreted into its equivalent of wakeful thought, is a psychic phenomenon for which no explanation is available. Every family of Riviera Presbyterian Church has been asked to read the Bible and pray together daily during National Christian Family Week and to undertake one project in which all members of the family participate. ** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element. * Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ). ** Every infinite game in which is a Borel subset of Baire space is determined. Every telephone company, whether large or small, determines its own ANAC for each individual central office, which tends to perpetuate the current situation of a mess of overlapping and / or spotty areas of coverage. : Every non-empty set A contains an element B which is disjoint from A. Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice. Every year since 1972 the BVI has hosted the Spring Regatta, which is a seven-day collection of sailing races throughout the islands. * 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 character is automatically continuous from A to C, since the kernel of a character is a maximal ideal, which is closed. Every computer contains an internal clock that regulates the rate at which instructions are executed and synchronizes all the various computer components. * Duality: Every statement, theorem, or definition in category theory has a dual which is essentially obtained by " reversing all the arrows ". Every context-sensitive grammar which does not generate the empty string can be transformed into an equivalent one in Kuroda normal form. According to Every, one example may be " the myth of St. George " and other stories about saints battling dragons, which were " modelled no doubt in many cases on older representations of the creator and preserver of the world in combat with chaos ". Every argues that " the disparagement of myth in our own civilization " stems partly from objections to perceived idolatry, objections which intensified in the Reformation, both among Protestants and among Catholics reacting against the classical mythology revived during the Renaissance. Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one which is in Chomsky normal form. Every four years, during which an extra 24 hours have accumulated, one extra day is added to keep the count coordinated with the sun's apparent position. Every node has a location, which is a number between 0 and 1. Every sequence can, thus, be read in three reading frames, each of which will produce a different amino acid sequence ( in the given example, Gly-Lys-Pro, Gly-Asn, or Glu-Thr, respectively ). Every time the boy had an injury which caused him internal or external bleeding, the Tsarina called on Rasputin, and the Tsarevich subsequently got better. Every week before June 19, the strip focuses on Garfield's birthday, which he dreads because of his fear of getting older. ::: Every Bill which shall have passed the House of Representatives and the Senate, shall, before it become a Law, be presented to the President of the United States ; If he approves he shall sign it, but if not he shall return it, with his Objections to that House in which it shall have originated ...

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