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Every finitely generated ideal of a Boolean ring is principal ( indeed, ( x, y )=( x + y + xy )).
Some Related Sentences
Every and finitely
Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.
# Every finitely generated ideal of A is principal ( i. e., A is a Bézout domain ) and A satisfies the ascending chain condition on principal ideals.
* Every finitely generated group with a recursively enumerable presentation and insoluble word problem is a subgroup of a finitely presented group with insoluble word problem
Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.
* 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 finitely presented group is recursively presented, but there are recursively presented groups that cannot be finitely presented.
Every one of the infinitely many vertices of G can be reached from v < sub > 1 </ sub > with a simple path, and each such path must start with one of the finitely many vertices adjacent to v < sub > 1 </ sub >.
" I can now state the physical version of the Church-Turing principle: ' Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means.
* Every substructure is the union of its finitely generated substructures ; hence Sub ( A ) is an algebraic lattice.
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.
This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of ( finitely or infinitely many ) copies of this module K < sup > n </ sup >.
Every ( bounded ) convex polytope is the image of a simplex, as every point is a convex combination of the ( finitely many ) vertices.
Every and generated
Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by
Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients.
Every film today, whether it be live-action, computer generated, or traditional hand-drawn animation is made up of hundreds of individual shots that are all placed together during editing to form the single film that is viewed by the audience.
A large part of the interest generated was due to the eclectic mix of samples and influences, evident on Every Man and Woman, which was dedicated to Dewey Bunnell of the band America.
Every ticket generated by the system has persistence or " history " showing what happened to the ticket within its life cycle.
Every profile also captures some non-crime statistics: the amount of overtime generated by members of the command, the number of department vehicle accidents, absence rates due to sick time and line-of-duty injuries, and the number of civilian complaints lodged against members of the unit.
Every and ideal
* 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.
* 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 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 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 field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index.
In Leedskalnin's own publication A Book in Every Home he implies his " Sweet Sixteen " was more an ideal than a reality.
Every meal can be eaten without any cooking ; when circumstances permit, the ideal method of preparation is to cook the entrees in a pressure cooker, heated on the standard issue Coleman stove, or by simply boiling the rations in its package in water.
Every free spirit looked upon her as protectoress and ideal .... Marguerite was the embodiment of charity.