A ring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that: (R, +) is an abelian group with identity element 0: (a + b) + c = a + (b + c) a + b = b + a 0 + a = a + 0 = a ∀a ∃(−a) such that a + −a = −a + a = 0 (R, ·) is a monoid with identit ...

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Ring mathematics, Ring mathematics - Formal definition, Ring mathematics - Alternative definitions, Ring mathematics - Examples, Ring mathematics - Basic theorems, Ring mathematics - Constructing new rings from given ones

Read more here: » Ring mathematics: Encyclopedia II - Ring mathematics - Formal definition

A ring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that: (R, +) is an abelian group with identity element 0: (a + b) + c = a + (b + c) a + b = b + a 0 + a = a + 0 = a ∀a ∃(−a) such that a + −a = −a + a = 0 (R, ·) is a monoid with identit ...

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Ring mathematics, Ring mathematics - Formal definition, Ring mathematics - Alternative definitions, Ring mathematics - Examples, Ring mathematics - Basic theorems, Ring mathematics - Constructing new rings from given ones, Ring mathematics - Categorical description

Read more here: » Ring mathematics: Encyclopedia II - Ring mathematics - Formal definition

In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. An antiautomorphism is an antihomomorphism that is a bijection from an object to itself. In group theory, an antihomomorphism is a map between two groups that reverses the order of multiplication. So if φ : X → Y is a group antihomomorphism, φ(xy) = φ(y)φ(x) for all x,y in X. The map that sends x to x ...

Including:

• Antihomomorphism - Examples

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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 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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In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). Congruence relation - Modular arithmetic. The prototypical example is modular arithmetic: for n a positive natural number, two integers a and b are called congruent modulo n if a − b is divisible by n. If and , then and . This turns the equivalence (mod ...

Including:

• Congruence relation - Modular arithmetic
• Congruence relation - Linear algebra
• Congruence relation - Universal algebra
• Congruence relation - Group theory
• Congruence relation - Ring theory
• Congruence relation - General case of kernels

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For an electrical switch that periodically reverses the current see commutator (electric) In mathematics, the commutator gives an indication of how poorly a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Commutator - Group theory. The commutator of two elements g and h of a group G is the element [g, h] = g−1h−1gh Including:

• Commutator - Group theory
• Commutator - Identities
• Commutator - Ring theory
• Commutator - Identities

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Ring - a circle. Finger ring, a circular band made to be worn on the finger, often precious Toe ring Arm ring, typically worn by men around the upper arm A piece of jewelry which can be fasted through the skin: e.g.: earring, nose ring, tongue ring, nipple ring, and other body piercings Collectible rings in video games

Including:
• Ring - Ring refers to...
• Ring - a circle
• Ring - a sound
• Ring - Things entitled ring or rings
• Ring - Things with ring or rings in the name or title

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In mathematics, an algebraic variety is essentially a set of common zeroes of a set of polynomials. Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry. Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a polynomial in one variable over the complex numbers is determined by the set of its roots, which is an inherently geometric object. Building on this result, Hilbert's Nullstellensatz provides a fun ...

Including:

• Algebraic variety - Formal definitions
• Algebraic variety - Affine varieties
• Algebraic variety - Projective varieties
• Algebraic variety - Basic results
• Algebraic variety - Discussion and generalizations

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In mathematics, an involution, or an involutary function, is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. Involution - General properties. The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other exa ...

Including:

• Involution - General properties
• Involution - Involutions in Euclidean geometry
• Involution - Involutions in differential geometry
• Involution - Involutions in ring theory
• Involution - Involutions in group theory

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From the axioms, one can immediately deduce that, for all elements a and b of a ring, we have 0a = a0 = 0 (−1)a = −a (−a)b = a(−b) = −(ab) (ab)−1 = b−1 a−1 if both a and b are invertible Other basic theorems The identity element 1 is unique If the ring has at least two elements then 0 ≠ 1 The bi ...

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Ring mathematics, Ring mathematics - Formal definition, Ring mathematics - Alternative definitions, Ring mathematics - Examples, Ring mathematics - Basic theorems, Ring mathematics - Constructing new rings from given ones

Read more here: » Ring mathematics: Encyclopedia II - Ring mathematics - Basic theorems

From the axioms, one can immediately deduce that, for all elements a and b of a ring, we have 0a = a0 = 0 (−1)a = −a (−a)b = a(−b) = −(ab) (ab)−1 = b−1 a−1 if both a and b are invertible Other basic theorems The identity element 1 is unique If the ring has at least two elements then 0 ≠ 1 The bi ...

See also:

Ring mathematics, Ring mathematics - Formal definition, Ring mathematics - Alternative definitions, Ring mathematics - Examples, Ring mathematics - Basic theorems, Ring mathematics - Constructing new rings from given ones, Ring mathematics - Categorical description

Read more here: » Ring mathematics: Encyclopedia II - Ring mathematics - Basic theorems

Adjoint functors - Ubiquity of adjoint functors. The idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory, it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those faced with giving tidy, systematic presentations of the subject would have noticed relations such as Hom(F(X), Y< ...

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Adjoint functors, 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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If G is a semidirect product of N and H, then the map φ : H → Aut(N) (where Aut(N) denotes the group of all automorphisms of N) defined by φ(h)(n) = hnh–1 for all h in H and n in N is a group homomorphism. Together N, H and φ determine G up to isomorphism, as we now show. Given any two groups N and H (not necessarily subgroups of a given group) and a group homomorphism φ : See also:

Semidirect product, Semidirect product - Some equivalent definitions, Semidirect product - Elementary facts and caveats, Semidirect product - Outer semidirect products, Semidirect product - Examples, Semidirect product - Relation to direct products, Semidirect product - Generalizations, Semidirect product - Notation

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Two integers a, b are said to be congruent modulo n if their difference is a multiple of n. In this case, we write a ≡ b (mod n). For instance, 38 ≡ 14 (mod 12) because 38 − 14 = 24 which is a multiple of 12. For positive numbers, congruence can also be thought of as asserting that two numbers have the same remainder after dividing by the modulus n. So, 38 ≡ 14 (mod 12) < ...

See also:

Modular arithmetic, Modular arithmetic - The congruence relation, Modular arithmetic - The ring of congruence classes, Modular arithmetic - Remainders, Modular arithmetic - Applications

Read more here: » Modular arithmetic: Encyclopedia II - Modular arithmetic - The congruence relation

The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other examples include include circle inversion, the ROT13 transformation, and the Beaufort polyalphabetic cipher. An involution is a kind of bijection. ...

See also:

Involution, Involution - General properties, Involution - Involutions in Euclidean geometry, Involution - Involutions in differential geometry, Involution - Involutions in ring theory, Involution - Involutions in group theory

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The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of primes in a unique way, i.e. unique except for the order. Primes are thus the "basic building blocks" of the natural numbers (The proof of this is below). For example, we can write and any other such factorization of 23244 will be identical except for the order of the factors. See prime fac ...

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Prime number, Prime number - Representing natural numbers as products of primes, Prime number - How many prime numbers are there?, Prime number - Finding prime numbers, Prime number - Some properties of primes, Prime number - Open questions, Prime number - The largest known prime, Prime number - Applications, Prime number - Primality tests, Prime number - Some special types of primes, Prime number - Prime gaps, Prime number - Formulae yielding prime numbers, Prime number - Generalizations, Prime number - Prime elements in rings, Prime number - Prime ideals, Prime number - Primes in valuation theory, Prime number - Quotes, Prime number - Primes in pop culture

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Let a and b be in the radical of an ideal I. Then, for some positive integers m and n, an and bm are in I. We will show that a + b is in I. Use the binomial theorem to expand (a+b)n+m−1: For each i, exactly one of the following conditions will hold: iSee also:

Radical of an ideal, Radical of an ideal - Definition, Radical of an ideal - Examples, Radical of an ideal - Proof that the radical is an ideal, Radical of an ideal - The nilradical of a ring, Radical of an ideal - Jacobson radicals, Radical of an ideal - Properties, Radical of an ideal - Applications

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Uploading a key to a key server reduces the level of security that can be expected from the key. The reduction is minor, but not insignificant. Because key servers are used to distribute keys which are part of key pairs used in public key cryptography, posting one of the key pairs can allow an adversary to perform types of cryptanalysis attacks which would not have been possible without it. Primarily, it enables known plaintext analysis. But, perhaps more importantly, in the case of public key pairs which make use of the RSA algorithm, posti ...

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Key server cryptographic, Key server cryptographic - History, Key server cryptographic - Public versus private keyservers, Key server cryptographic - Privacy concerns, Key server cryptographic - Weaken security, Key server cryptographic - Key server etiquette

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Ring - a circle. Finger ring, a circular band made to be worn on the finger, often precious Toe ring Arm ring, typically worn by men around the upper arm A piece of jewelry which can be fasted through the skin: e.g.: earring, nose ring, tongue ring, nipple ring, and other body piercings Collectible rings in video games e.g.: Rings (Legend of Zelda), Rings (Soni ...

See also:

Ring, Ring - Ring refers to..., Ring - a circle, Ring - a sound, Ring - Things entitled ring or rings, Ring - Things with ring or rings in the name or title

Read more here: » Ring: Encyclopedia II - Ring - Ring refers to...