Page "Prime ideal" ¶ 9
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any and ring
One means to help the birds occurs to me: Let the chimes that ring over Washington Square twice daily, discontinue any piece of music but one.
The morning hawk, hungry for any eatable, killable, digestible item, kept his eyes on the ring of anchored ships that lay off the shores in the bay, sheltered by the Jersey inlets.
`` The reason you are in the ring today is to show your ability to present to any judge the most attractive picture of your dog that the skillful use of your aids can produce.
Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.
The same definition holds in any unital ring or algebra where a is any invertible element.
* Any ring A is an algebra over its center Z ( A ), or over any subring of its center.
* Any commutative ring R is an algebra over itself, or any subring of R.
The projective plane over any alternative division ring is a Moufang plane.
Bragi generously offers his sword, horse, and an arm ring as peace gift but Loki only responds by accusing Bragi of cowardice, of being the most afraid to fight of any of the Æsir and Elves within the hall.
If a fighter is knocked down during the fight, determined by whether the boxer touches the canvas floor of the ring with any part of their body other than the feet as a result of the opponent's punch and not a slip, as determined by the referee, the referee begins counting until the fighter returns to his or her feet and can continue.
The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.
For the case of a non-commutative base ring R and a right module M < sub > R </ sub > and a left module < sub > R </ sub > N, we can define a bilinear map, where T is an abelian group, such that for any n in N, is a group homomorphism, and for any m in M, is a group homomorphism too, and which also satisfies
* β-Lactams not fused to any other ring are named monobactams.
( valid for any elements x, y of a commutative ring ),
As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain ( PID ).
* The spectrum of any commutative ring with the Zariski topology ( that is, the set of all prime ideals ) is compact, but never Hausdorff ( except in trivial cases ).
Mail armour provided an effective defence against slashing blows by an edged weapon and penetration by thrusting and piercing weapons ; in fact a study conducted at the Royal Armouries at Leeds concluded that " it is almost impossible to penetrate using any conventional medieval weapon " Generally speaking, mail's resistance to weapons is determined by four factors: linkage type ( riveted, butted, or welded ), material used ( iron versus bronze or steel ), weave density ( a tighter weave needs a thinner weapon to surpass ), and ring thickness ( generally ranging from 18 to 14 gauge in most examples ).
* The reinforce: This portion of the piece is frequently divided into a first reinforce and a second reinforce, but in any case is marked as separate from the chase by the presence of a narrow circular reinforce ring or band at its foremost end.
Although most often used for matrices whose entries are real or complex numbers, the definition of the determinant only involves addition, subtraction and multiplication, and so it can be defined for square matrices with entries taken from any commutative ring.
This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements.
Especially, the fact that the integers and any polynomial ring in one variable over a field are Euclidean domains such that the Euclidean division is easily computable is of basic importance in computer algebra.
Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

any and R
For United States expenditures under subsections ( A ), ( B ), ( D ), ( E ), ( F ), ( H ) through ( R ) of Section 104 of the Act or under any of such subsections, the rupee equivalent of \$200 million.
Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that ( x, y, z ) belongs to R.
So, for example, while R < sup > n </ sup > is a Banach space with respect to any norm defined on it, it is only a Hilbert space with respect to the Euclidean norm.
In simpler term, Biotechnology is the research and development in the laboratory that involves bioinformatics for exploration, extraction, exploitation and production from any living organisms and any source of biomass by means of biochemical engineering where high value-added products could be planned ( reproduced by Biosynthesis, for example ), fore-casted, formulated, developed, manufactured and marketed for the purpose of sustainable operations ( for the return from bottomless initial investment on R & D ) and gaining durable patents rights ( for exclusives rights for sales, and prior to this to receive national and international approval from the results on animal experiment and human experiment, especially on the pharmaceutical branch of biotechnology to prevent any undetected side-effects on safety concerns by using the products ), for more about the biotechnology industry, see.
From Comrade Semichastny's speech I learn that the government, ' would not put any obstacles in the way of my departure from the U. S. S. R .' For me this is impossible.
* In the cocountable topology on R ( or any uncountable set for that matter ), no infinite set is compact.
For any subset A of Euclidean space R < sup > n </ sup >, the following are equivalent:
The space Q < sub > p </ sub > of p-adic numbers is complete for any prime number p. This space completes Q with the p-adic metric in the same way that R completes Q with the usual metric.
This is a generalization of the Heine – Borel theorem, which states that any closed and bounded subspace S of R < sup > n </ sup > is compact and therefore complete.
Likewise, the problem of computing a quantity on a manifold which is invariant under differentiable mappings is inherently global, since any local invariant will be trivial in the sense that it is already exhibited in the topology of R < sup > n </ sup >.
Democide is a term revived and redefined by the political scientist R. J. Rummel as " the murder of any person or people by a government, including genocide, politicide, and mass murder.
Although siblinghood is symmetric ( if A is a sibling of B, then B is a sibling of A ) and transitive on any 3 distinct people ( if A is a sibling of B and C is a sibling of B, then A is a sibling of C, provided A is not C ( Note that " is a sibling of " is NOT a transitive relation, since A R B, and B R A implies A R A by transitivity )), it is not reflexive ( A cannot be a sibling of A ).

any and maximal
** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.
The elements 2 and 1 + √(− 3 ) are two " maximal common divisors " ( i. e. any common divisor which is a multiple of 2 is associated to 2, the same holds for 1 + √(− 3 )), but they are not associated, so there is no greatest common divisor of a and b.
It states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.
The Hausdorff maximal principle states that, in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.
Here a maximal totally-ordered subset is one that, if enlarged in any way, does not remain totally ordered.
The holographic principle was inspired by black hole thermodynamics, which implies that the maximal entropy in any region scales with the radius squared, and not cubed as might be expected.
For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε ; that is, it is always possible to transmit with arbitrarily small block error.
For example, the direct sum of the R < sub > i </ sub > form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime.
Since all maximal paths have the same number of black nodes, by property 5, this shows that no path is more than twice as long as any other path.
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.
For general groups, Cauchy's theorem guarantees the existence of an element, and hence of a cyclic subgroup, of order any prime dividing the group order ; Sylow's theorem extends this to the existence of a subgroup of order equal to the maximal power of any prime dividing the group order.
If r is the degree of the primitive generator polynomial, then the maximal total block length is, and the associated code is able to detect any single-bit or double-bit errors.
For any normal modal logic L, a Kripke model ( called the canonical model ) can be constructed, which validates precisely the theorems of L, by an adaptation of the standard technique of using maximal consistent sets as models.
Given a ring R and a proper ideal I of R ( that is I ≠ R ), I is a maximal ideal of R if any of the following equivalent conditions hold:
The converse is not always true: for example, in any nonfield integral domain the zero ideal is a prime ideal which is not maximal.
For an R module A, a maximal submodule M of A is a submodule M ≠ A for which for any other submodule N, if M ⊆ N ⊆ A then N = M or N = A.
A subset T is totally ordered if for any s, t in T we have s ≤ t or t ≤ s. Such a set T has an upper bound u in P if t ≤ u for all t in T. Note that u is an element of P but need not be an element of T. An element m of P is called a maximal element ( or non-dominated ) if there is no element x in P for which m < x.
In any human age group there is however considerable variation in maximal pupil size.
As a rule of thumb, any sort of construction that takes as input a fairly general object ( often of an algebraic, or topological-algebraic nature ) and outputs a compact space is likely to use Tychonoff: e. g., the Gelfand space of maximal ideals of a commutative C * algebra, the Stone space of maximal ideals of a Boolean algebra, and the Berkovich spectrum of a commutative Banach ring.

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