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Page "Associative algebra" ¶ 38
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Any and ring
* Any ring A can be considered as a Z-algebra in a unique 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 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 characteristic
Any mutations for the preference opposite to the given characteristic, though tending to promote survival against ecological selection, nevertheless tend not to survive in the gene pool because male offspring that result from matings based on the preference are less sexually attractive to the majority of the females in the population, and thus infrequently chosen as mates.
Any characteristic of a Class III game makes it a Class III game.
Limiting: Any process by which a specified characteristic ( usually amplitude ) of the output of a device is prevented from exceeding a predetermined value.
Note that for orientable compact surfaces without boundary, the Euler characteristic equals, where is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and counts the number of handles.
Any convex polyhedron's surface has Euler characteristic
Any electromagnetic radiation consists of an electric field component E and a magnetic field component H. In the far-field, the relationship between the electric field component E and the magnetic component H is that characteristic of any freely propagating wave, where ( in units where c = 1 ) E is equal to H at any point in space.
*" Any form of cultural knowledge or social activity characteristic of a social group and / or cultural group that can be recognized by other groups such as Western anthropologists, but not necessarily by the group of origin, as mathematical knowledge or mathematical activity ".
Any definite conclusion to Cynewulf being either Northumbrian or Mercian has been hard to come by, but linguistic evidence suggests that the medial e in the signed Cynewulf would have, during the broad window period of Cynewulf ’ s existence, been characteristic of a Mercian dialect.
Any monic polynomial is the characteristic polynomial of some matrix.
Any language, in Bakhtin's view, stratifies into many voices: " social dialects, characteristic group behavior, professional jargons, generic languages, languages of generations and age groups, tendentious languages, languages of the authorities, of various circles and of passing fashions.
:; Cavity back: Any iron whose design characteristic is such that the weight is distributed primarily around the outer edges of the club-head in order to maximize forgiveness on off-center hits.
Any semiconductor diode has an exponential voltage / current characteristic that gives an effective " knee " voltage sometimes used as a voltage reference.
Any protons ejected from the target itself were identified by their characteristic energies.

Any and n
Any subset of R < sup > n </ sup > ( with its subspace topology ) that is homeomorphic to another open subset of R < sup > n </ sup > is itself open.
Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of orthogonal group O ( n ) by choosing the origin to be a fixed point.
Any n × n permutation matrix can be constructed as a product of no more than n − 1 transpositions.
Any orthogonal matrix of size n × n can be constructed as a product of at most n such reflections.
Any rotation matrix of size n × n can be constructed as a product of at most n ( n − 1 )/ 2 such rotations.
Any n × n matrix A of the form
To show that U ( n ) is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal.
# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.
for some p between 1 and n. Any two such expressions of q ( at the same point m of M ) will have the same number p of positive signs.
* Any positive multiple of the repunit R < sub > n </ sub >< sup >( b )</ sup > contains at least n nonzero digits in base b.
Any covering space of a differentiable manifold may be equipped with a ( natural ) differentiable structure that turns p ( the covering map in question ) into a local diffeomorphism – a map with constant rank n.
Any n × n real symmetric matrix A determines a quadratic form q < sub > A </ sub > in n variables by the formula

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