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Page "Associative algebra" ¶ 39
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Any and ring
* Any ring A can be considered as a Z-algebra in a unique way.
* Any ring of characteristic n is a ( Z / nZ )- algebra in the same way.
* Any commutative ring R is an algebra over itself, or any subring of R.
* Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication.
* Any discrete valuation ring.
Any legal wrestler is open to attack from any direction at any time, including when they are downed, as long as they are within the ring area enclosed by the ring ropes.
Any type or number of topologies may be used -- star, bus, ring, etc.
* Any ring, thought of as a category with only one object, is a preadditive category.
Any ( non-graded ) ring A can be given a gradation by letting A < sub > 0 </ sub >
Any number system that forms a commutative ring — for instance, the integers, the real numbers, and the complex numbers — can be extended to a wheel in which division by zero is always possible ; however, in such a case, " division " has a slightly different meaning.
Any ring can be seen as a preadditive category with a single object.
* Any commutative ring becomes a *- ring with the trivial involution.
* Any ring is also a semiring.
* Any ring R can be considered as a one-object preadditive category ; the category of left modules over R is the same as the additive functor category Add ( R, Ab ) ( where Ab denotes the category of abelian groups ), and the category of right R-modules is Add ( R < sup > op </ sup >, Ab ).
Any quotient of a ring by a maximal ideal is a simple ring.
Any prime number p gives rise to an ideal pO < sub > K </ sub > in the ring of integers O < sub > K </ sub > of a quadratic field K.
* Any algebra over a commutative ring K may be regarded as a purely even superalgebra over K ; that is, by taking A < sub > 1 </ sub > to be trivial.
Any n-dimensional formal group law gives an n dimensional Lie algebra over the ring R, defined in terms of the quadratic part F < sub > 2 </ sub > of the formal group law.
* Any two-sided ideal of a ring R is an R-bimodule.
* Any module over a commutative ring R is automatically a bimodule.

Any and is
Any organism that falters or misperceives the signals or weakens is done.
Any information we have here in Taiwan is at your disposal ''.
Any claimant whose claim is denied, or is approved for less than the full amount of such claim, shall be entitled, under such regulations as the Commission may prescribe, to a hearing before the Commission, or its duly authorized representatives, with respect to such claim.
Any alteration of one of these factors is distortion, although we generally use that word only for effects so pronounced that they can be stated quantitatively on the basis of standard tests.
Any musician playing Beethoven here, where Beethoven was born, is likely to examine his own interpretations with special care.
Any action is understood to create " seeds " in the mind that sprout into the appropriate results ( Pāli vipaka ) when they meet the right conditions.
** Any union of countably many countable sets is itself countable.
Any information may be conveyed by an analog signal ; often such a signal is a measured response to changes in physical phenomena, such as sound, light, temperature, position, or pressure.
Any quantity of energy absorbed by the electron in excess of this amount is converted to kinetic energy according to the conservation of energy.
Any breach of these laws is a criminal offence and can lead to fine of up to £ 5, 000 or six months in prison.
The prohibition against retaliation or coercion applies broadly to any individual or entity that seeks to prevent an individual from exercising his or her rights or to retaliate against him or her for having exercised those rights ... Any form of retaliation or coercion, including threats, intimidation, or interference, is prohibited if it is intended to interfere.
Any word or phrase that exactly reproduces the letters in another order is an anagram.
Any work ( δW ) done must be done at the expense of internal energy U, since no heat δQ is being supplied from the surroundings.
" Any threat of war Clovis could make would only be effective if they were neighbors ; " it is nowhere written that Syagrius was handed over in 486 or 487.
Any player may declare the game over at any time during his turn if either of two conditions is true: one chain has 41 or more tiles, or there is at least one chain on the board and every chain on the board has 11 or more tiles.
Any person who is seen as a threat to civil aviation is banned from flying.
Any form of intelligent life would do ; so, specifying carbon-based life, per se, is irrelevant.
* Any base that is water-soluble and forms hydroxide ions or the solution of a base in water.

Any and algebra
Any Banach algebra ( whether it has an identity element or not ) can be embedded isometrically into a unital Banach algebra so as to form a closed ideal of.
Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra.
* Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra.
Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.
* Any Lie group G defines an associated real Lie algebra.
Any algebra whose elements are idempotent is also power-associative.
:“ Any linear map from V to an algebra A can be uniquely extended to an algebra homomorphism from T ( V ) to A .”
A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ ( y )= 0 for all x, y in g. Any weight on a Lie algebra g vanishes on the derived algebra and hence descends to a weight on the abelian Lie algebra g /.
Any associative algebra A over the field K becomes a Lie algebra over K with the Lie bracket:
: Any linear transformation f: V → A from V to an algebra A over K can be uniquely extended to an algebra homomorphism from T ( V ) to A as indicated by the following commutative diagram:
* Any Z or N-graded algebra may be regarded as superalgebra by reading the grading modulo 2.
Any affine group scheme is the spectrum of a commutative Hopf algebra ( over a base S, this is given by the relative spectrum of an O < sub > S </ sub >- algebra ).

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