* 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 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 .”

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 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 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 threat of war Clovis could make would only be effective if they were neighbors ; " it is nowhere written that Syagrius was handed over in 486 or 487.

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.

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 and all profits made from book deals or any movies about Lindh's experience will be automatically handed over to the federal government.

Any list of these would have to include the Night of the Living Houses ( said to be the first comic strip to enter the collection of the Louvre ) wherein Nemo and a friend are chased down a city street by a gang of tenement houses on legs ; the Walking Bed, in which Nemo and Flip ride over the rooftops on the increasingly long limbs of Nemo's bed ( see illustration ); and the Befuddle Hall sequence, wherein Nemo and his friends attempt to find their way out of a funhouse environment of a Beaux-Arts interior turned topsy-turvy.

Any Hohenstaufen in Sicily was bound to have claims over the cities of Lombardy, and as a check to Manfred, Urban IV introduced Charles of Anjou into the equation to place the crown of the Two Sicilies in the hands of a monarch amenable to papal control.

Any component of target velocity perpendicular to the line of sight cannot be determined by using the Doppler effect alone, but it can be determined by tracking the target's azimuth over time.

In July 2006, a book, ' Peace at Any Price: How the World Failed Kosovo ', written by two former senior staffers at UNMIK, exposed a catalogue of errors and incompetence in the institution over its seven year history.

# Any relatively high-bandwidth transmission over a short period.

Any Transport over MPLS-Cisco Systems.

Any large body of water upwind will impact lake-effect precipitation to the lee of a downwind lake by adding moisture or pre-existing lake-effect bands, which can re-intensify over the downwind lake.

Any production over 100 g must be notified to the OPCW and a country can have a stockpile of no more than one tonne of these chemicals.

Any person over the age of majority and of sound mind ( having appropriate mental capacity ) can draft his own will with or without the aid of a lawyer.

Any quadratic polynomial over the complex numbers ( polynomials of the form where,, and ∈ ) can be factored into an expression with the form using the quadratic formula.

Any amount over and above that required to maintain the capital at the beginning of the period is profit.

Any number of transputers could be connected together over even longish links ( tens of metres ) to form a single computing " farm ".

Any member of the priesthood who presides over a congregation can, and often is, known as " pastor " or ( if an elder ), " presiding elder ".

Any two adjacent conductors can be considered a capacitor, although the capacitance will be small unless the conductors are close together for long distances or over a large area.

Any police entry of an individual's home always requires a warrant ( for either search or arrest ), absent exigent circumstances, or the free and voluntary consent of a person with reasonably apparent use of or control over the property.

# Any sequence satisfying the recurrence relation can be written uniquely as a linear combination of solutions constructed in part 1 as λ varies over all distinct roots of p ( t ).

* 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 commutative ring becomes a *- ring with the trivial involution.

* Any bounded, distributive lattice is a commutative, idempotent semiring under join and meet.

* 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 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 ).