In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, then a×b=b×a. The study of commutative rings is called commutative algebra. Commutative ring - Examples. The most important example is the ring of integers with the two operations of addition and multiplication. Ordinary multiplication of inte ...

Including:

• Commutative ring - Examples
• Commutative ring - Constructing new commutative rings from given ones
• Commutative ring - General discussion

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Commensurability mathematics - Commensurability in general. Generally, two quantities are commensurable if both can be measured in the same units. For example, a distance measured in miles and a quantity of water measured in gallons are incommensurable. A time measured in weeks and a time measured in minutes are commensurable because a week is a constant number of minutes (10080), so that one can convert between the two units by multiplying or dividing by 10080. Commensurabilit ...

Including:

• Commensurability mathematics - Commensurability in general
• Commensurability mathematics - Commensurability in mathematics

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Conan the Barbarian (also known as Conan the Cimmerian, from the name of his homeland, Cimmeria) is a literary character created by Robert E. Howard in a series of fantasy pulp stories published in Weird Tales in the 1930s. Conan the Barbarian - Setting. The Conan stories take place on Earth, but in the mythical (created by Howard) "Hyborian Age," between the time of the sinking of Atlantis and the rise of the known ancient civilizations. According to Howard himself: "...between the years w ...

Including:

• Conan the Barbarian - Setting
• Conan the Barbarian - Characteristics
• Conan the Barbarian - Appearance
• Conan the Barbarian - Abilities and Characteristics
• Conan the Barbarian - Influences
• Conan the Barbarian - The Original Robert E. Howard Conan Stories
• Conan the Barbarian - Conan stories published in Weird Tales
• Conan the Barbarian - Conan stories by Howard not published in his lifetime
• Conan the Barbarian - Unfinished Conan stories by Howard
• Conan the Barbarian - Other Conan related material by Howard
• Conan the Barbarian - Textual history
• Conan the Barbarian - Book editions
• Conan the Barbarian - The Gnome Press editions 1959-1957
• Conan the Barbarian - The Lancer/Ace paperback editions 1966-1977
• Conan the Barbarian - The Donald M. Grant editions 1974-1989
• Conan the Barbarian - The Berkeley editions 1977
• Conan the Barbarian - The Bantam editions 1978-1982
• Conan the Barbarian - The Ace Maroto editions 1978-1981
• Conan the Barbarian - The Tor editions 1982-2004
• Conan the Barbarian - The Gollancz editions 2000-2001
• Conan the Barbarian - The Wandering Star/Del Rey editions 2003-2005
• Conan the Barbarian - Other media
• Conan the Barbarian - Movies
• Conan the Barbarian - TV Series
• Conan the Barbarian - Cartoons
• Conan the Barbarian - Comics
• Conan the Barbarian - Games
• Conan the Barbarian - Parody and other references
• Conan the Barbarian - Characters
• Conan the Barbarian - Quotes
• Conan the Barbarian - Quotes from Howard's original Conan stories
• Conan the Barbarian - Conan the Barbarian movie quotes

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In computer science, a datatype (often simply a type) is a name or label for a set of values and some operations which one can perform on that set of values. Programming languages implicitly or explicitly support one or more datatypes; these types may act as a statically or dynamically checked constraint, ensuring valid programs for a given language. Datatype - Basis. Assigning datatypes ("typing") has the basic purpose of giving some semantic meaning to otherwise meaningless collections of bits. Typ ...

Including:

• Datatype - Basis
• Datatype - Type checking
• Datatype - Static and dynamic typing
• Datatype - Static and dynamic type checking in practice
• Datatype - Strong and weak typing
• Datatype - Polymorphism and types
• Datatype - Explicit or implicit declaration and inference
• Datatype - Collections of types
• Datatype - Specialized types
• Datatype - Compatibility equivalence and substitutability

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In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts of mathematics, like products and inverse limits. Accordingly, the dual notion of a colimit, generalizes disjoint unions and direct sums. Limits and colimits have strong relationships to the categorial concepts of universal morphisms and adjoint functors. Limit category theory - Definition. Before defining limits, it is useful to defin ...

Including:

• Limit category theory - Definition
• Limit category theory - Examples
• Limit category theory - Complete categories
• Limit category theory - Continuous functors
• Limit category theory - Colimits
• Limit category theory - Creation of Limits and Co-Limits

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D is an object-oriented, imperative systems programming language designed by Walter Bright of Digital Mars as a successor to C++. He has done this by adding some features and reducing the complexity of C++ syntax. Examples of successors to C++ include Java and C#. D extends C++ by implementing design by contract, unit testing, true modules, automatic memory management (garbage collection), first class arrays, associative arrays, dynamic arrays, array slicing, nested functions, closures (anonymous functions), and has a reenginee ...

Including:

• D programming language - D examples
• D programming language - Example 1
• D programming language - Example 2

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Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find som ...

Including:

• Algebraic geometry - Zeroes of simultaneous polynomials
• Algebraic geometry - Affine varieties
• Algebraic geometry - Regular functions
• Algebraic geometry - The category of affine varieties
• Algebraic geometry - Projective space
• Algebraic geometry - The modern viewpoint
• Algebraic geometry - Notes and history

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In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves enable one to discuss in a refined way what is a local property, as appl ...

Including:

• Sheaf mathematics - Introduction
• Sheaf mathematics - The formal definition
• Sheaf mathematics - Definition of a presheaf
• Sheaf mathematics - The gluing axiom
• Sheaf mathematics - Examples
• Sheaf mathematics - Morphisms of sheaves
• Sheaf mathematics - Stalks of a sheaf at a point and germs of functions
• Sheaf mathematics - The étale space of a sheaf
• Sheaf mathematics - Generalizations
• Sheaf mathematics - History

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A software emulator allows computer programs to run on a platform (computer architecture and/or operating system) other than the one for which they were originally written. Unlike a simulation, which only attempts to reproduce a program's behavior, an emulation attempts to precisely model the state of the device being emulated. A popular use of emulators is to mimic the experience of running arcade games or console games on Linux, Mac OS X, and Microsoft Windows. Emulating these on modern desktop computers is usually less cumbe ...

Including:

• Emulator - Structure
• Emulator - Memory subsystem
• Emulator - CPU simulator
• Emulator - I/O

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In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. Such functors are ubiquitous in mathematics. Adjoint functors are studied in a branch of mathematics known as category theory. Like much of category theory, the general notion of adjoint functors arises at an abstract level beyond the everyday usage of mathematicians. Adjoint functors can be considered from several different points of view. This article starts with a number of introductory sections considering some ...

Including:

• Adjoint functors - Motivation
• Adjoint functors - Ubiquity of adjoint functors
• Adjoint functors - Deep problems formulated with adjoint functors
• Adjoint functors - Adjoint functors as solving optimization problems
• Adjoint functors - The case of partial orders
• Adjoint functors - Formal definitions
• Adjoint functors - Examples
• Adjoint functors - Properties
• Adjoint functors - Uniqueness of adjoints
• Adjoint functors - Relation to universal constructions
• Adjoint functors - Characterization via unit and co-unit
• Adjoint functors - Adjoints preserve certain limits
• Adjoint functors - Additivity
• Adjoint functors - Composition
• Adjoint functors - Adjoint pairs extend equivalences
• Adjoint functors - General existence theorem

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In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have nice properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. Abelian category - Definitions. A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal. By a theorem of Pe ...

Including:

• Abelian category - Definitions
• Abelian category - Examples
• Abelian category - Elementary properties
• Abelian category - Related concepts
• Abelian category - History

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Cherry Corporation is a US company best known for its computer keyboards. They also manufacture a large range of products including sensors, input devices and automotive modules. Cherry Corporation has three divisions:- Cherry Automotive Cherry Electrical Products Cherry GmbH ...

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In category theory, a zero morphism is a special kind of "trivial" morphism. Suppose C is a category, and for any two objects X and Y in C we are given a morphism 0XY : X → Y with the following property: for any two morphism f : R → S and g : U → V we obtain a commutative diagram: Then the morphisms 0XY are c ...

Including:

• Zero morphism - Examples

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In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. In other words, the order of elements in a product doesn't matter. Such groups are generally easier to understand. Abelian groups are named after Niels Henrik Abel. Groups that are not commutative are called non-abelian (rather than non-commutative). Abelian group - NotationIncluding:

• Abelian group - Notation
• Abelian group - Examples
• Abelian group - Multiplication table
• Abelian group - Properties
• Abelian group - Finite abelian groups
• Abelian group - List of small abelian groups
• Abelian group - Relation to other mathematical topics
• Abelian group - A note on the typography

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Abstract algebra is the field of mathematics concerned with the study of algebraic structures called groups, rings and fields. These structures were defined formally in the nineteenth century, and, indeed, the study of abstract algebra was motivated by the need for more rigor in mathematics. The study of abstract algebra has brought into full view intricacies of the logical assumptions on which the whole of mathematics and natural science is built, and today there is scarcely a branch of mathematics which doesn't utilize the results o ...

Including:

• Abstract algebra - History and examples
• Abstract algebra - An example

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A category C is given by two pieces of data: a class of objects and a class of morphisms. There are two operations defined on every morphism, the domain (or source) and the codomain (or target). Morphisms are often depicted as arrows from their domain to their codomain, e.g. if a morphism f has domain X and codomain Y, it is denoted f : X → Y. The set of all morphisms from X to Y is denoted homC(X,Y) or simply hom(X, Y). (Some authors write MorC(X ...

See also:

Morphism, Morphism - Definition, Morphism - Types of morphisms, Morphism - Examples

Read more here: » Morphism: Encyclopedia II - Morphism - Definition

Monster Manual - Early Dungeons & Dragons. The earliest Dungeons and Dragons box games did not have separate Monster Manuals but contained the monsters in the manuals in the boxed set. The original boxed set included a Book 2: Monsters and Treasure. After the publication of Advanced Dungeons & Dragons, the game was still published in level-based boxes. Monsters of the appropriate levels were included in the rulebooks in Basic/Expert/Companion/Ma ...

See also:

Monster Manual, Monster Manual - Current Monster Manual 3rd Edition Dungeons & Dragons, Monster Manual - Earlier Monster Manuals, Monster Manual - Early Dungeons & Dragons, Monster Manual - Advanced Dungeons & Dragons 1st Edition, Monster Manual - 2nd Edition Advanced Dungeons & Dragons

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A completely analogous definition holds for the case of linear representations of G. Specifically, if X and Y are two linear representations of G then a linear map f : X → Y is called an intertwiner of the representations if it commutes with the action of G. Alternatively, an intertwiner for representations of G over a field K is the same thing as a module homomorphism of K[G]-modules, where K< ...

See also:

Equivariant, Equivariant - Intertwiners, Equivariant - Categorical description

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The main purpose of the invariant basis number condition is that modules over any ring satisfying the IBN satisfy an analogue of the dimension theorem for vector spaces: any two bases for a module over an IBN ring have the same cardinality. Assuming the ultrafilter lemma (a strictly weaker form of the axiom of choice), this result is actually equivalent to the definition given here, and can be taken as an alternative definition. The rank of a free module Rn over an IBN ring R is defined to be the ...

See also:

Invariant basis number, Invariant basis number - Definition, Invariant basis number - Discussion, Invariant basis number - Examples, Invariant basis number - Other results

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In topological language, we try to find covers of unfamiliar objects. Finding a generator of an abelian category allows one to express every object as a quotient of a direct sum of copies of the generator. Finding a cogenerator allows one to express every object as a subobject of a direct product of copies of the cogenerator. One is often interested in projective generators (even finitely generated projective generators, called progenerators) and minimal injective cogene ...

See also:

Injective cogenerator, Injective cogenerator - The abelian group case, Injective cogenerator - General theory, Injective cogenerator - In general topology

Read more here: » Injective cogenerator: Encyclopedia II - Injective cogenerator - General theory