* 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 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 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 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 Z or N-graded algebra may be regarded as superalgebra by reading the grading modulo 2.

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 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 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 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 configuration of charges or currents has a ' dipole moment ', which describes the dipole whose field is the best approximation, at large distances, to that of the given configuration.

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 collection of objects and morphisms defines a ( possibly large ) directed graph G. If we let J be the free category generated by G, there is a universal diagram F: J → C whose image contains G. The limit ( or colimit ) of this diagram is the same as the limit ( or colimit ) of the original collection of objects and morphisms.

Any class can be viewed as a category whose only morphisms are the identity morphisms.

: Any magistrate, or in the absence of any magistrate any commissioned officer in Her Majesty's Naval, Military or Air Force Service or any police officer above the rank of inspector, in whose view a riot is being committed, or who apprehends that a riot is about to be committed by persons being assembled within his view, may make or cause to be made a proclamation in the Queen's name, in such form as he thinks fit, commanding the rioters or persons so assembled to disperse peaceably.

Any sort of radio transmitter whose presence is suspected is detectable with suitable equipment.

Any source whose appearance or behavior indicates that he is willing to talk should be noted by the screeners.

Any conjectures are largely based on the date of her father's death ( 1006 ), the approximate date of her betrothal to Cnut ( 1004 × 1016, see below ) and the time by which she had borne him sons, whose ages are themselves difficult to establish.

Any timelike curve admits a comoving observer whose " time axis " corresponds to that curve, and, since no observer is privileged, we can always find a local coordinate system in which lightcones are inclined at 45 degrees to the time axis.

Any oscillator whose oscillations are driven by a system that almost always is dissipating energy may be called a relaxation oscillator.

Any misuse of the arm by the employee of the company in whose custody the arm is placed is the responsibility of the company / agency.

" The Amerasian Foundation ( AF ) and Amerasian Family Finder ( AFF ) define an Amerasian as: " Any person who was fathered by a citizen of the United States ( an American servicemen, American expatriate, or U. S. Government Employee ( Regular or Contract )) and whose mother is, or was, an Asian National Asian.

Any physical object whose radius R becomes less than or equal to the Schwarzschild radius will undergo gravitational collapse and become a black hole.

** " Common variation ": Any player whose score hits exactly " 45 " is set back to Zero — The dreaded Forty-Five Rule.

Any man whose name appeared on the list was ipso facto stripped of his citizenship and excluded from all protection under law ; reward money was given to any informer who gave information leading to the death of a proscribed man, and any person who killed a proscribed man was entitled to keep part of his estate ( the remainder went to the state ).

-- Any attempt to access a member whose existence depends

Any node whose lower bound is greater than can be discarded.

Any human with courage and presence of mind can outwit a troll, and those whose faith is strong can even challenge them to mortal combat.

Any Iraqi whose citizenship was withdrawn for political, religious, racial, or sectarian reasons has the right to reclaim his Iraqi citizenship, and each Iraqi is guaranteed the right to carry more than one citizenship.

An introductory title screen – repeated as the Foreword to the 1946 novelization by Eric Warman – contains an explicit statement: " This is the story of two worlds, the one we know and another which exists only in the mind of a young airman whose life and imagination have been violently shaped by war ", adding " Any resemblance to any other world known or unknown is purely coincidental ".

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 pair of successive shapes in such a sequence is a skew shape whose diagram contains at most one box in each column ; such shapes are called horizontal strips.

Any tragedy, he maintains, has six elements: plot, character, and thought ( the objects of imitation ), diction and melody ( the means of imitation ), and spectacle ( the manner of imitation ).

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 normal subgroup has a corresponding quotient group, formed from the larger group by eliminating the distinction between elements of the subgroup.

* Any two finite fields with the same number of elements are isomorphic.

Any racial elements from these influences have not remained part of the Anthroposophical Society as contemporary adherents of the society have either not adopted or repudiated these beliefs.

Any tampering with the original image of a film, particularly to crop it to fit a television screen, implies a compromise of the original image, and the cropping of a widescreen image to a full screen image for standard televisions requires skill by a film editor to prevent undue loss of elements of the composition.

Any building requires a certain amount of internal infrastructure to function, which includes such elements like heating / cooling, power and telecommunications, water and wastewater etc.

Any choice of basis will have p elements, which are the dimensionless variables.

* Any partially ordered set can be interpreted as a category: the objects are the elements of, and there is a single morphism from to if and only if.

Any additional periods are expected to contain a larger number of elements than the seventh period, as they are calculated to have an additional so-called g-block, containing 18 elements with partially filled g-orbitals in each period.

Any corrosion products of gold or platinum tend to decompose spontaneously into pure metal, which is why these elements can be found in metallic form on Earth and have long been valued.

Any electrical network can be analysed as multiple, interconnected electrical elements in a schematic diagram or circuit diagram, each of which affects the voltage in the network or current through the network.

Any formed faculty does not cease to exist on the removal of its stimulus ; in virtue of its fundamental property, tenacity, it sinks back as a trace ( Spur ) into unconsciousness, whence it may be recalled by the application to it of another stimulus, or by the attraction towards it of some of the movable elements or newly formed original powers.

Any elements that decohere from each other via environmental interactions are said to be quantum entangled with the environment.

Any member who by word or action showed that his principles were contrary to the constitution and the rights of man was to be expelled, a rule which later on facilitated the " purification " of the society by the expulsion of its more moderate elements.

# Any assignment of the elements of a formal language to natural numbers in such a way that the numbers can be manipulated by an algorithm to simulate manipulation of elements of the formal language.

Any philosophy that asserts two elements such as the yin-yang of Chinese philosophy will also look for a term to reconcile the two, to ensure that both belong to the same sphere of discourse.

Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into k chains then the width of the order must be at most k. Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width.