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Page "Simple ring" ¶ 2
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Any and quotient
Any normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.
* Any quotient of a diffeological space is a diffeological space.
Any such surface is the Kummer variety of the Jacobian of a smooth hyperelliptic curve of genus 2 ; i. e. a quotient of the Jacobian by the Kummer involution x ↦ − x.
* Any irreducible representation R of H gives an irreducible representation of G using the quotient map from G to H ( that is, as a restricted representation ).
Any quotient group of a group with property FA has property FA.
* Any subspace and any quotient space by a closed subspace of a nuclear space is nuclear.

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 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 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 by
Any fee so determined shall be entered as a part of such award, and payment thereof shall be made by the Secretary of the Treasury by deducting the amount thereof from the total amount paid pursuant to the award.
Any and all payments required to be made by the Secretary of the Treasury under this title pursuant to any award made by the Commission to the Government of the United States shall be covered into the Treasury to the credit of miscellaneous receipts.
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 free elections that were to be held in Poland would have to produce a government in which Moscow had complete confidence, and all pressure from the West for free voting by anti-Soviet elements in Poland would be met by restrictions on voting by these elements.
Any example of grand or exquisite diction may have been created by the poet who compiled numerous lays into the two works we possess or may be due to one of his completely unknown fellow-craftsmen.
Any animal could when travelin' fast, be sent heels over head by the simple process of overtakin' the brute, seizin' its tail, and givin' the latter a pull to one side.
Any needy family living in San Francisco can obtain toys by writing to Christmas Toys, 676 Howard street, San Francisco 5, and listing the parent's name and address and the age and sex of each child in the family between the ages of 1 and 12.
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 individual has one of six possible genotypes ( AA, AO, BB, BO, AB, and OO ) that produce one of four possible phenotypes: " A " ( produced by AA homozygous and AO heterozygous genotypes ), " B " ( produced by BB homozygous and BO heterozygous genotypes ), " AB " heterozygotes, and " O " homozygotes.
* Any Time Murder, a reference to the role performed by Meka Srikanth in the film Shankar Dada MBBS
Any individual alignment could indicate a direction by chance, but he planned to show that together the distribution of alignments was non-random, showing that there was an astronomical intent to the orientation of at least some of the alignments.
Any node in the data structure can be reached by starting at root node and repeatedly following references to either the left or right child.
Any number of players may be replaced by substitutes during the course of the game.
Any other craters that might have formed in the Tethys Ocean would have been obscured by tectonic events like the relentless northward drift of Africa and India.
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 locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification.
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 non-Jewish slave bought by a Jew will be confiscated by the state ; if a Jew attempts to circumcise a non-Jewish slave, the slave will be freed and the Jew shall face capital punishment ; any Christian slaves owned by a Jew will be taken away and freed ;

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