Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.

Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows.

* Any ( pre ) abelian category is finitely complete and finitely cocomplete.

Any cyclotomic extension ( for either definition ) is abelian.

Any abelian variety is isogenous to a product of simple abelian varieties.

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 finite abelian group, with the discrete topology.

* 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 semisimple Lie algebra or abelian Lie algebra is a fortiori reductive.

The theorem states: Any linearly ordered abelian group can be embedded as an ordered subgroup of the additive group ℝ < sup > Ω </ sup > endowed with a lexicographical order, where ℝ is the additive group of real numbers ( with its standard order ), and Ω is the set of Archimedean equivalence classes of.

Any abelian group can be turned into a zero ring by defining the product of any two elements to be 0.

Reflecting Schleicher ′ s reputation for deviousness and being untrustworthy, Hermann Göring joked in 1932 :" Any Chancellor who has Herr von Schleicher on his side must expect sooner or later to be sunk by the Schleicher torpedo, there was a joke current in political circles -" General von Schleicher ought really to have been an Admiral for his military genius lies in shooting under water at his political friends "".

Additional stage credits after Bonanza include Two for the Seesaw, A Thousand Clowns, One Flew Over the Cuckoo's Nest, Any Wednesday and The Sound of Music ( as Captain von Trapp ).

Any advantages for the Austrians however were canceled out by the crushing defeat on 3 July at Königgrätz, where Ludwig von Benedek was surprised by the speed of Helmuth von Moltke's concentric advance into Bohemia.

Any chromaticity difference between the source and reference illuminants were to be abridged with a von Kries-type chromatic adaptation transform.

Any of the stages V < sub > α </ sub > and L < sub > α </ sub > leading to the construction of the von Neumann universe V and Gödel's constructible universe L are transitive sets.

In the 1949 talk, Von Neumann famously quipped that, " Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin.

* 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 Lie group G defines an associated real Lie algebra.

Any algebra whose elements are idempotent is also power-associative.

:“ Any linear map from V to an algebra A can be uniquely extended to an algebra homomorphism from T ( V ) to A .”

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 data that will be sent is serialized ( also referred to as marshalling in CORBA speak ) using the built-in QDataStream operators available in all of the Qt classes.

Any bounded operator L that has finite rank is a compact operator ; indeed, the class of compact operators is a natural generalisation of the class of finite-rank operators in an infinite-dimensional setting.

Any plans to reopen the island to visitors will require the participation of the Veracruz Reef System Park, to which it belongs, as well as Secretariat of the Navy, INAH, the Veracruz Aquarium and other government agencies negotiating with the local associations of hotels, tour operators and fishermen.

Any user-defined conversion must be explicitly marked as explicit or implicit, unlike C ++ copy constructors and conversion operators, which are both implicit by default.

Any alteration of one of these factors is distortion, although we generally use that word only for effects so pronounced that they can be stated quantitatively on the basis of standard tests.

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 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 decline in amphibian numbers will have an impact on the patterns of predation.

Any historical material on anagrams must always be interpreted in terms of the assumptions and spellings that were current for the language in question.

Any player may declare the game over at any time during his turn if either of two conditions is true: one chain has 41 or more tiles, or there is at least one chain on the board and every chain on the board has 11 or more tiles.

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 alcohol ( beer, wine, spirits ) cannot be bought on Sundays before noon.

Any player is usually allowed to control or bet in as many boxes as desired at a single table, but it is prohibited for an individual to play on more than one table at a time or to place multiple bets within a single box.

Any measure μ defined on the σ-algebra of Borel sets is called a Borel measure.

Any desired arbitration logic is placed on the daughter cards.

Any attempt to impose an organization on them is bound to be somewhat arbitrary.

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

The Environmental Modification Convention ( ENMOD ), formally the Convention on the Prohibition of Military or Any Other Hostile Use of Environmental Modification Techniques is an international treaty prohibiting the military or other hostile use of environmental modification techniques.

One episode in Series 5 of Steptoe and Son was entitled " Any Old Iron ", for the same reason, when Albert thinks that Harold is ' on the turn '.

Any piece of DNA with the point centromere DNA sequence on it will typically form a centomere if present in the appropriate species.

Any person who felt himself wronged might lay an information before the Council of Areopagus, on declaring what law was broken by the wrong done to him.

Any elected or appointed board, committee, town officer or ten voters, may place an article on the warrant.

Any rain falling on the above-water rim quickly percolates through the surface sand and encounters the salt water underneath.

* 1976 – The United Nations General Assembly adopts the Convention on the Prohibition of Military or Any Other Hostile Use of Environmental Modification Techniques.

Any player on the kicking team behind the kicker, and including the kicker, can recover the kick.

:* Any total order on X has exactly one equivalence class, X itself, because x ~ y for all x and y ;