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Page "*-algebra" ¶ 24
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Any and commutative
* 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 commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if and only if there exists z such that x + z = y.
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 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 bounded, distributive lattice is a commutative, idempotent semiring under join and meet.
* 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 module over a commutative ring R is automatically a bimodule.
Any commutative semigroup is a medial magma, and a medial magma has an identity element if and only if it is a commutative monoid.
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 ).
Any complete group variety ( variety here meaning reduced and geometrically irreducible separated scheme of finite type over a field ) is automatically commutative, by an argument involving the action of conjugation on jet spaces of the identity.
Any commutative algebra is a supercommutative algebra if given the trivial gradation ( i. e. all elements are even ).

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 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 ring can be seen as a preadditive category with a single object.
* 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 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 and becomes
* Any vector space V endowed with the identically zero Lie bracket becomes a Lie algebra.
Any amendment so ratified becomes a valid part of the Constitution, provided that no state " shall be deprived of its equal Suffrage in the senate ," without its consent.
# Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.
Any handmade envelope is effectively a lettersheet because prior to the folding stage it offers the opportunity for writing a message on that area of the sheet that after folding becomes the inside of the face of the envelope.
* Any piece, save for the king, may be captured and removed from the board if it becomes surrounded on two opposite sides by enemies.
Any subsequent child born to the couple becomes the half-sister or half-brother of the prior children from previous relations.
Any physical object whose radius R becomes less than or equal to the Schwarzschild radius will undergo gravitational collapse and become a black hole.
Any dish with huitlacoche must include a slow simmer of the fungus until it becomes black which also removes most of the starch of the corn and what is left is a black oily paste.
Any subsystem that becomes excessively loaded can compromise the performance of other clients contending for that subsystem.
Any associative algebra A over the field K becomes a Lie algebra over K with the Lie bracket:
Any anisotropic or isotropic linear algebraic group over a field becomes split over the algebraic closure, so this distinction is interesting from the point of view of Algebraic number theory.
* Any U. S. financial institution that becomes aware that it has possession of or control over funds in which a designated FTO or its agent has an interest must retain possession of or control over the funds and report the funds to the Office of Foreign Assets Control of the U. S. Department of the Treasury.
" Any large organization, Michels pointed out, has to create a bureaucracy in order to maintain its efficiency as it becomes larger — many decisions have to be made daily that cannot be made by large numbers of disorganized people.
Any object she sits on or lies upon during this period becomes a midras l ' tumah ( carrier of tumah ).
Any skew semistandard tableau of shape / with positive integer entries gives rise to a sequence of partitions ( or Young diagrams ), by starting with, and taking for the partition places further in the sequence the one whose diagram is obtained from that of by adding all the boxes that contain a value ≤ in ; this partition eventually becomes equal to.
Any time a new release is added or old release edited, that entry becomes flagged as needing " votes " ( initially, " review ," but this term caused confusion ).
Any Taelon who becomes disconnected from the Commonality ( which has happened on at least one occasion ) reverts back to an Atavus, with the reappearance of the dormant Shaqarava.
Any device with its Bluetooth connection turned on and set to " discoverable " ( able to be found by other Bluetooth devices in range ) may be susceptible to Bluejacking, and possibly to Bluesnarfing when, and if, Bluesnarfing of the current Bluetooth security becomes possible.
Any point of location becomes frequented / enlivened / vivified not merely by the passing by visits but when people gather and stay there for a duration.
Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.
Any child born to a married woman, even if she is known to have been unfaithful, is presumed to be her husband's, unless she is so promiscuous that such a presumption becomes unsupportable ,( Shulhan Arukh Even haEzer 4: 15 ) or if she enters a public relationship with another man ( 4: 16 ).
Any flat wheel that becomes too dangerous to travel on ( a big flat spot on the train wheel ) will be counted as a defect.
Any ace that becomes available is put onto the foundations, to be built up by suit.

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