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

: 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:

A graph coloring is an assignment of one of k colors to a graph G so that the endpoints of each edge have different colors, for some number k. Any coloring corresponds to a homomorphism from G to a complete graph K < sub > k </ sub >: the vertices of K < sub > k </ sub > correspond to the colors of G, and f maps each vertex of G with color c to the vertex of K < sub > k </ sub > that corresponds to c. This is a valid homomorphism because the endpoints of each edge of G are mapped to distinct vertices of K < sub > k </ sub >, and every two distinct vertices of K < sub > k </ sub > are connected by an edge, so every edge in G is mapped to an adjacent pair of vertices in K < sub > k </ sub >.

# Any finite-dimensional unital associative algebra A has a natural homomorphism to its own endomorphism ring End ( A ).

Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j if and only if i ≤ j.

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.

* quantification Any morphism f: X → Y in a category with pullbacks induces a monotonous map acting by pullbacks ( A monotonous map is a functor if we consider the preorders as categories ).

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 transformation of the plane leaving this pattern invariant can be decomposed into a translation, ( x, y ) → ( n + x, y ), optionally followed by a reflection in either the horizontal axis, ( x, y ) → ( x ,− y ), or the vertical axis, ( x, y ) → (− x, y ), provided that this axis is chosen through or midway between two dots, or a rotation by 180 °, ( x, y ) → (− x ,− y ) ( ditto ).

# Any string after an M may be completely duplicated ( Mx → Mxx )

# Any three consecutive I's ( III ) may be replaced with a single U ( xIIIy → xUy )

# Any two consecutive U's may be removed ( xUUy → xy )

* Any two maps f, g: Y → X are homotopic.

* Any map f: Y → X is null-homotopic.

Any individual has one of six possible genotypes ( AA, AO, BB, BO, AB, and OO ) that produce one of four possible phenotypes: " A " ( produced by AA homozygous and AO heterozygous genotypes ), " B " ( produced by BB homozygous and BO heterozygous genotypes ), " AB " heterozygotes, and " O " homozygotes.

Any subset of B is also well-ordered.

Brown continued releasing recordings: in 1993, he issued the album, Universal James, which included Brown's final Billboard charted single, " Can't Get Any Harder ", which peaked at # 76 on the US R & B chart and # 59 on the UK chart.

The blitter's block copying mode takes zero to three data sources in memory, called A, B and C, performs a programmable boolean function on the data sources and writes the result to a destination area, D. Any of these four areas can overlap.

Any such function is called a Urysohn function for A and B.

Any member of B is in the set ker r + im u ; since for all b in B, b =

Any attempts to make this schedule serial would give off two different results ( either T1's version of A / B is shown, or T2's version of A and B is shown ), and would not be the same as the above schedule.

Any prospect for further mining of the lead-zinc resource would now require significant investment to bring in mining equipment, and it would need to come entirely by road unless a B. C .- Alaska railway is built or the White Pass route is reopened to freight traffic to Whitehorse.

Gates 20-28 are swing gates between Concourse B and C. Any US bound flight is treated as a " C " Gate, whereas any Canadian-bound / International-bound flight is treated as a " B " Gate.

Robert Christgau's review of Into the Unknown was more favorable than that of How Could Hell Be Any Worse ?, which received a B rating, whereas Into the Unknown was given a A-rating.

In early June, Banks released the second single off the album, " Any Girl " featuring R & B singer Lloyd.

Any pa ( or sentential logic ) formula B can be viewed as an ordered tree with branches.

# Any three points A, B, C of a plane α, which do not lie in the same straight line, completely determine that plane.

Any four points A, B, C, D of a straight line can always be so arranged that B shall lie between A and C and also between A and D, and, furthermore, that C shall lie between A and D and also between B and D.

Any genus zero handlebody is homeomorphic to the three-ball B < sup > 3 </ sup >.

( B ) RISK ATTRIBUTABLE TO FAULT OF EITHER PARTY: Any damage or loss which arises as a result of the fault or neglect of the seller or the buyer or their respective agents as the case may be shall be borne by that party at fault.

Any case of obstruction falls into one of two categories generally called type " A " and type " B ", after the subsection of Official Baseball Rule 7. 06, in which each type of obstruction is described.