ideals

For instance, let y be a root of y2 + y + 6 = 0, then the ring of integers of the field is , which means all a + by with a and b integers form the ring of integers. An example of a nonprincipal ideal in this ring is 2a + yb with a and b integers; the cube of this ideal is principal, and in fact the class group is cyclic of order three. The corresponding class field is obtained by adjoining an element w satisfying w3 - w - ...

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Ideal number, Ideal number - Example, Ideal number - History

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There are a number of conditions which are equivalent to a and b being coprime: There exist integers x and y such that ax + by = 1 (see Bézout's identity). The integer b has a multiplicative inverse modulo a: there exists an integer y such that by ≡ 1 (mod a). In other words, b is a unit in the ring Z/aZSee also:

Coprime, Coprime - Properties, Coprime - Cross notation group, Coprime - Generalizations

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Rings of polynomials over fields have many special properties; properties that follow from the fact that polynomial rings are not, in some sense, "too large". Emmy Noether first discovered that the key property of polynomial rings is the ascending chain condition on ideals. Noetherian rings are named after her. For noncommutative rings, we must distinguish between three very similar concepts: A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals. A ring is right-Noetheri ...

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Noetherian ring, Noetherian ring - Introduction, Noetherian ring - Characterizations of Noetherian rings, Noetherian ring - Uses of Noetherian rings, Noetherian ring - Examples, Noetherian ring - Properties

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A semiring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that: (R, +) is a commutative monoid with identity element 0: (a + b) + c = a + (b + c) 0 + a = a + 0 = a a + b = b + a (R, ·) is a monoid with identity element 1: (a·b)·c = a·(b·c) 1·a ...

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Semiring, Semiring - Definition, Semiring - Examples, Semiring - Semiring theory, Semiring - Further generalizations

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A partially ordered set (L, ≤) is a complete lattice if every subset A of L has both a greatest lower bound (infimum, meet) and a least upper bound (supremum, join). These are denoted by: A (meet) and A (join). Note that in the special case where A is the empty set the meet of A will be the greatest element of L. Likewise, the join of the empty set yields the least element. Since the definition also assures the existence of binary meets and joins, complete latt ...

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Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

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One can then check that the set of all polynomials with coefficients in the ring R, together with the addition + and the multiplication mentioned above, forms itself a ring, the polynomial ring over R, which is denoted by R[X]. Formally these two ring operations are functions defined on with values in R[X], given by the formulas and If ...

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Polynomial ring, Polynomial ring - Definition of a polynomial, Polynomial ring - The polynomial ring R[X], Polynomial ring - The polynomial ring in several variables, Polynomial ring - Equivalent definition, Polynomial ring - Properties, Polynomial ring - Some uses of polynomial rings

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Intuitively, the Boolean prime ideal theorem states that there are "enough" prime ideals in a Boolean algebra in the sense that we can extend every ideal to a maximal one. This is of practical importance for proving Stone's representation theorem for Boolean algebras, a special case of Stone duality, in which one equips the set of all prime ideals with a certain topology and can indeed regain the original Boolean algebra (up to isomorphism) from this data. Furthermore, it turns out that in applications one can freely choose either to ...

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Boolean prime ideal theorem, Boolean prime ideal theorem - Prime ideal theorems, Boolean prime ideal theorem - Boolean prime ideal theorem, Boolean prime ideal theorem - Further prime ideal theorems, Boolean prime ideal theorem - Applications, Boolean prime ideal theorem - Literature

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Let C be a category, and let c be an object of C. A sieve S on c is a subfunctor of Hom(−, c), i.e., for all objects c′ of C, S(c′) ⊆ Hom(−, c), and for all arrows f:c″→c′, S(f) is the restriction of Hom(f, c), the pullback by f, to S(c′). Put another way, a sieve is a collection S of arrows with a common codomain which satisfies the functoriality condition, ...

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Sieve category theory, Sieve category theory - Definition, Sieve category theory - Pullback of sieves, Sieve category theory - Properties of sieves

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In real analysis, a polynomial is a certain type of a function of one or several variables (see polynomial), or in other words, a polynomial function. This definition cannot be adapted to a general ring, however. For example, over the ring Z/2Z of integers modulo 2, the polynomial P(X)=X2+X=X(X+1) takes only the value 0, as when k is an integer, k(k+1) is always even. But we would expec ...

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Polynomial ring, Polynomial ring - Definition of a polynomial, Polynomial ring - The polynomial ring R[X], Polynomial ring - The polynomial ring in several variables, Polynomial ring - Equivalent definition, Polynomial ring - Properties, Polynomial ring - Some uses of polynomial rings

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Sun Tzu Thucydides The ancient Greek historian Thucydides, who wrote the History of the Peloponnesian War, is also cited as an intellectual forebearer of realpolitik. Machiavelli One of the most famous proponents was Niccolò Machiavelli, best known for his Il Principe (The Prince) (pb.1532). Machiavelli held that the sole aim of a prince was to seek power, regardless of religious or ethical considerations. Richelieu Machiavelli's ideas were further expanded and implemented by Cardinal Richelieu and his ...

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Realism in international relations, Realism in international relations - Basic theory, Realism in international relations - History of realism, Realism in international relations - Modern realism, Realism in international relations - Classical realism, Realism in international relations - Structural or Neo-realism, Realism in international relations - Modern realist statesmen, Realism in international relations - Criticisms of realism

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In social studies, a political ideology is a certain ethical, set of ideals, principles, doctrines, myths or symbols of a social movement, institution, class, or large group that explain how society should work, and offer some political and cultural blueprint for a certain social order. A political ideology largely concerns itself with how to allocate power and to what ends it should be used. It can be a construct of political thought, often defining political parties and their policy. Studies of the concept of ideology itself (rather than specific ideologies) have been c ...

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Ideology, Ideology - Ideology in everyday society, Ideology - History of the concept of ideology, Ideology - The analysis of ideology, Ideology - Ideology as an instrument of social reproduction, Ideology - Louis Althusser's Ideological State Apparatuses, Ideology - Feminism as critique of ideology, Ideology - Political ideologies, Ideology - List of political ideologies, Ideology - Epistemological ideologies

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As clearly evidenced by literature around the world, poor body images have existed at least since the widespread availability of mirrors, but one of the reasons most often cited for this continuing body dissatisfaction among young women is modern media influence, including that from movies, television, and magazines. Media representatives often reply that they are merely reflecting the ideals of the current generation or using whatever image best sells their products. However, research has shown that the media play a large role in reinforcin ...

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Body image, Body image - Research: Measuring Body Image, Body image - Causes and Influences, Body image - Relationship to Psychological Disorders, Body image - Attractiveness and Social Issues, Body image - Information on Specific Minority Populations

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A polynomial function in one real variable can be represented by a graph. The graph of the zero polynomial f(x) = 0 is the x-axis. The graph of a degree 0 polynomial f(x) = a0 , where a0 ≠ 0, is a horizontal line with y-intercept a0 The graph of a degree 1 polynomial (or linear function) f(x) = a0 + a1x , whe ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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The Noetherian property is central in ring theory and in areas that make heavy use of rings, such as algebraic geometry. The reason behind this is that the Noetherian property is in some sense the ring-theoretic analogue of finiteness. For example, the Noetherian-ness of polynomial rings over a field allows us to prove that any infinite set of polynomial equations can be replaced with a finite set with the same solutions. As another application, we mention Krull's principal ideal theorem: Every principal ideal in a commutative Noetherian ring has height one. This early result was the first to suggest ...

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Noetherian ring, Noetherian ring - Introduction, Noetherian ring - Characterizations of Noetherian rings, Noetherian ring - Uses of Noetherian rings, Noetherian ring - Examples, Noetherian ring - Properties

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"What is the meaning of life?" is a question many people ask themselves at some point during their lives. Some people believe that the meaning of life is one or more of the following: Survival and temporal success ...to accumulate wealth and increase social status ...to compete or co-operate with others ...to destroy others who harm you, or to practice nonviolence and nonresistance ...to gain and exercise power ...to leave a legacy, such as a work of art or a book ...to ...

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Meaning of life, Meaning of life - Popular beliefs, Meaning of life - Scientific approaches and theories, Meaning of life - Philosophical views, Meaning of life - Value as meaning, Meaning of life - Atheist views, Meaning of life - Existentialist views, Meaning of life - Humanist views, Meaning of life - Nihilist views, Meaning of life - Positivist views, Meaning of life - Pragmatist views, Meaning of life - Transhumanist views, Meaning of life - Religious beliefs, Meaning of life - Spiritual views, Meaning of life - Humorous treatments, Meaning of life - General philosophy topics, Meaning of life - General philosophy lists

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There are other, equivalent, definitions for a ring R to be left-Noetherian: Every left ideal I in R is finitely generated, i.e. there exist elements a1, ..., an in I such that I = Ra1 + ... + Ran. Every non-empty set of left ideals of R has a maximal element with respect to set i ...

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Noetherian ring, Noetherian ring - Introduction, Noetherian ring - Characterizations of Noetherian rings, Noetherian ring - Uses of Noetherian rings, Noetherian ring - Examples, Noetherian ring - Properties

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Some researchers also found that men judge the female figure they found most attractive as heavier than women's ratings of the ideal body shape. In contrast, that most women, including overweight women, desire men with a very low percentage of body fat, whether they be thin or muscular. This suggests that, contrary to the media focus, men are far more likely to be attracted to larger woman than women are to be attracted to larger men. Additionally, men are also more likely to be unsatisfied with their height, due to a perceived prefer ...

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Body image, Body image - Research: Measuring Body Image, Body image - Causes and Influences, Body image - Relationship to Psychological Disorders, Body image - Attractiveness and Social Issues, Body image - Information on Specific Minority Populations

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Poor body images can often contribute to the onset of a variety of eating disorders, including anorexia nervosa, bulimia, and binge eating disorder. Other possible effects of the cultural obsession with looking slender include excessive exercising, fad diets, and lawsuits involving fast food chains. Concerns with body image have been linked to a decrease in self esteem and an increase in dieting among young women. This latter trend has been identified as an indicator of the onset of eating disorders such as anorexia nervosa and bulimi ...

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Body image, Body image - Research: Measuring Body Image, Body image - Causes and Influences, Body image - Relationship to Psychological Disorders, Body image - Attractiveness and Social Issues, Body image - Information on Specific Minority Populations

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Realism in international relations - Classical realism. George F. Kennan - Containment Nicholas Spykman - Geostrategy, Containment Herman Kahn - Nuclear strategy Hans Morgenthau E.H. Carr Reinhold Niebuhr Arnold Wolfers Charles Beard Walter Lippmann Realism in international relations - Structural or Neo-realism. Neo-Realism resembles Classical Realism on most accounts. However, Neo-Realism pred ...

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Realism in international relations, Realism in international relations - Basic theory, Realism in international relations - History of realism, Realism in international relations - Modern realism, Realism in international relations - Classical realism, Realism in international relations - Structural or Neo-realism, Realism in international relations - Modern realist statesmen, Realism in international relations - Criticisms of realism

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The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. Furthermore, the appearance of hypercomplex numbers in the mid-nineteenth century undercut the pre-eminence of fields in mathematical analysis. Richard Dedekind introduced the concept of a ring. The term ring (Zahlring) was coined by David Hilbert in the article Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897. The first axiomatic definition of a ring was given by Adolf Fraenkel in an essay in Journal für die reine und angewandte Mathema ...

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Ring theory, Ring theory - History, Ring theory - Elementary introduction, Ring theory - Definition, Ring theory - Some useful theorems, Ring theory - Generalizations

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The isomorphism theorems are also valid for modules over a fixed ring R (and therefore also for vector spaces over a fixed field). One has to replace the term "group" by "R-module", "subgroup" and "normal subgroup" by "submodule", and "factor group" by "factor module". The isomorphism theorems are also valid for rings, ring homomorphisms and ideals. One has to replace the term "group" by "ring", "subgroup" and "normal s ...

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Isomorphism theorem, Isomorphism theorem - Groups, Isomorphism theorem - First isomorphism theorem, Isomorphism theorem - Second isomorphism theorem also known as the third isomorphism theorem, Isomorphism theorem - Third isomorphism theorem also known as the second isomorphism theorem, Isomorphism theorem - Rings and modules, Isomorphism theorem - General

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Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense. That is, if f: A → B is a morphism in a preadditive category, then the kernel of f is the equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism. Unlike with products and coproducts, the kernel and cokernel ...

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Preadditive category, Preadditive category - Examples, Preadditive category - Elementary properties, Preadditive category - Additive functors, Preadditive category - Biproducts, Preadditive category - Kernels and cokernels, Preadditive category - Special cases

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