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Page "Preadditive category" ¶ 7
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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 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 ( 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 and thought
Any lingering suspicion that this was a trick Al Budd had thought up was dispelled.
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 thought or idea expressed in a language, prepared in a form suitable for transmission by any means of communication.
Any object radiating sound, including a loudspeaker system, can be thought of as being composed of combinations of such simple point sources.
Though having been gone for a few episodes ( and thought to have been " retired "), Comic Book Guy made an appearance in an episode that aired later in the season, " Any Given Sundance ".
Any government that uses neoliberal economic thought for its right-wing policies can be effectively labelled neoliberal.
Any suggestion of these attributes and names being convinced of as separate is thought to entail polytheism.
Any other variation over the last 200 – 300 years is thought to be much smaller than this.
Any ordinal number can be made into a topological space by endowing it with the order topology ( since, being well-ordered, an ordinal is in particular totally ordered ): in the absence of indication to the contrary, it is always that order topology which is meant when an ordinal is thought of as a topological space.
Any alteration to these techniques is thought to hinder Scientology's effectiveness.
Any positive immunological effect is thought to be the result of melatonin acting on high affinity receptors ( MT1 and MT2 ) expressed in immunocompetent cells.
Any 16-bit drivers and programs would patch or hook what they thought was the real MS-DOS, but which was in reality just a decoy.
His novel Babylon (" Generation П " or " Generation P " is the Russian title ) bears on its cover the inscription, " Any thought that occurs in the process of reading this book is subject to copyright.
Any attempt to explain theoretical thought without acknowledging non-theoretical factors is destined to fall into irresolvable antinomies.
Any solar religion or solar symbolism were not yet established in any useful form and it is now thought that king Raneb might actually have been the first king who adopted extended religious thought about the sun and the sky.
Any attempt to describe a doctrine will reflect a particular viewpoint, and for that reason cannot be thought of as definitive.
" Any charges against me are essentially charges of thought crimes ," he said.

Any and category
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.
For 2009, the winner in the " least likely to succeed " category was " Any name of the decade 2000 – 2009, such as: Naughties, Aughties, Oughties, Pot stickers, etc.
:- Any agreement or category of agreements between undertakings
:- Any decision or category of decisions by associations of undertakings
:- Any concerted practice or category of concerted practices that improve the production or distribution of goods, or promotes technical or economic progress, while allowing consumers a fair share of the resulting benefit, and that does not:
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 preordered set ( P, ≤) forms a small category, where the objects are the members of P, the morphisms are arrows pointing from x to y when x ≤ y.
Any ordinal number can be seen as a category when viewed as an ordered set.
Any monoid ( any algebraic structure with a single associative binary operation and an identity element ) forms a small category with a single object x.
Any group can be seen as a category with a single object in which every morphism is invertible ( for every morphism f there is a morphism g that is both left and right inverse to f under composition ) by viewing the group as acting on itself by left multiplication.
Any directed graph generates a small category: the objects are the vertices of the graph, and the morphisms are the paths in the graph ( augmented with loops as needed ) where composition of morphisms is concatenation of paths.
* 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 ).
* 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 topological space can be viewed as a category by taking the open sets as objects, and a single morphism between two open sets and if and only if.
Any finite product in a preadditive category must also be a coproduct, and conversely.
Any category with this property is necessarily a thin category: for any two objects there can be at most one morphism from one object to the other.
* Any ( pre ) abelian category is finitely complete and finitely cocomplete.
# Any group G may be regarded as an " abstract " category with one object,, and one morphism for each element of the group.
Any directed poset can be considered as a small category where the morphisms consist of arrows if and only if.
Any Other Weapons ( AOWs )— this is a broad " catch-all " category used to regulate any number of firearms which the BATFE under the NFA enforces registration and taxation.

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