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Page "Associative algebra" ¶ 42
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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 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.
* 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 matrices
* Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.
* Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1's on the diagonal of some rank, and any finite dimensional irreducible representation of such a group is 1 dimensional.
Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices.
* Any square matrix with unit Euclidean norm is the average of two unitary matrices.
Any set of Hermitian matrices which obey these relations are allowed.
Any unipotent algebraic group is isomorphic to a closed subgroup of the group of upper triangular matrices with diagonal entries 1, and conversely any such subgroup is unipotent.
Any 2 × 2 matrix can be written in the form, where I is the identity matrix, σ < sub > i </ sub > are the Pauli matrices and u is a unit vector.

Any and with
Any posse riding down the street to demand Blue Throat's surrender would be wiped out with one deadly burst of fire.
Any abilities I may have were achieved in their present shape from experience in sharing in the growth and control of my business, coupled with raising my family.
Any attempt to reconcile this statement of the central issue in the campaign of 1956 with the nature of the man who could not conceive it as the central issue will at least resolve our confusions about the chaotic and misleading results of the earnestness of both doctors and President in a situation which should never have arisen.
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 irregularity on the piston heads will make it impossible, with normal means, to determine the final combustion chamber volume because the volume displaced by the piston heads cannot be readily computed.
Any reputable French interne can supply you with a dozen similar instances, and I'll presently recount a case out of my own personal experience, but, for the moment, let's resume our catalogue.
Any of the antithyroid drugs, of either type, if given in large enough doses for a long period of time will cause goitre, owing to inhibition of thyroid hormone synthesis, with production of hypothyroidism.
Any adjustments which are made, Mr. Grenier said earlier this month, will appear on the balance of the tax bill since most of the town's taxpayers take the option of paying quarterly with the balance due next year.
Any shareholder of any of these funds who finds a mistake in his account certainly should get in touch with the bank.
Any musician playing Beethoven here, where Beethoven was born, is likely to examine his own interpretations with special care.
Most of the C code being written nowadays is based on ANSI C. Any program written only in standard C and without any hardware dependent assumptions is virtually guaranteed to compile correctly on any platform with a conforming C implementation.
Any company incorporating Bluetooth wireless technology into products, using the technology to offer goods and services or simply re-branding a product with Bluetooth technology must become a member of the Bluetooth SIG.
Any codex of Bede's Easter cycle is normally found together with a codex of his " De Temporum Ratione ".
Love and Erlandson continued to pursue with the band, and released the single " Be A Man "— an outtake from the Celebrity Skin sessions — for the soundtrack of the Oliver Stone film Any Given Sunday ( 1999 ).
Any differential area with normal vector of a given internal surface area, bounding a portion of the body, experiences a contact force arising from the contact between both portions of the body on each side of, and it is given by
# Any collection of closed subsets of X with the finite intersection property has nonempty intersection.
Any season ticket holder could put themselves forward for election, with a certain number of nominations, and votes were cast by all season ticket holders over the age of 18.
Any pellets which are too small to be finished granules of couscous and fall through the sieve will be again rolled and sprinkled with dry semolina and rolled into pellets.
Any successes that the Nationalists did make, however, were met with constant political and military upheavals.
Any piece of DNA with the point centromere DNA sequence on it will typically form a centomere if present in the appropriate species.
# Any experience which is inconsistent with the organization of the structure of the self may be perceived as a threat, and the more of these perceptions there are, the more rigidly the self structure is organized to maintain itself.
Any rational number with a denominator whose only prime factors are 2 and / or 5 may be precisely expressed as a decimal fraction and has a finite decimal expansion.
Any units of angular measure can be used for declination, but it is customarily measured in the degrees ( ° ), minutes ( ' ), and seconds ( " ) of sexagesimal measure, with 90 ° equivalent to circle.
Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters.

0.698 seconds.