Ring theory - Definition. Formally, a ring is an abelian group (R, +), together with a second binary operation * such that for all a, b and c in R, a * (b * c) = (a * b) * c a * (b + c) = (a * b) + (a * c) (a + b) * c = ( ...

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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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Fundamental principles common to realist theories: The international system is anarchical. Sovereign states are the principal actors in the international system. Dismissal of the independent influence of international organizations, sub-state, or trans-state actors. Focus on the primary importance of nationalism, as opposed to sub-national groupings, or transnational ideological of cultural groupings. States are rational actors, acting in their national interest. Distrust ...

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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 most common operation on a sieve is pullback. Pulling back a sieve S on c by an arrow f:c′→c gives a new sieve f*S on c′. This new sieve consists of all the arrows in S which factor through c′. There are several equivalent ways of defining f*S. The simplest is: For any object d of C, f*S(d) = { gSee also:

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

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We will go over a typical application of Zorn's lemma: the proof that every ring R contains a maximal ideal. The set P here consists of all (two-sided) ideals in R, except R itself. This set is partially ordered by set inclusion. We are done if we can find a maximal element in P. The ideal R was excluded because maximal ideals by definition are not equal to R. We want to apply Zorn's lemma, and so we take a totally ordered subset T of P and have to show that T has a ...

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Zorn's lemma, Zorn's lemma - An example application, Zorn's lemma - Sketch of the proof of Zorn's lemma, Zorn's lemma - History

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First we state the isomorphism theorems for groups, where they take a simpler form and state important properties of quotient groups (also called factor groups). All three involve "modding out" by a normal subgroup. Isomorphism theorem - First isomorphism theorem. If G and H are groups and f is a homomorphism from G to H, then the kernel K of f is a normal subgroup of G, and the quotient group G/K is isomorphic to the image of f. If See also:

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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All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that includes only the operation of multiplication. Every polynomial in expanded form is a sum of terms (where subtraction is carried out by addition of negative numbers). Polynomials are classified by their degree and number of variables. The degree of a term in a polynomial is the su ...

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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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Meaning of life - Value as meaning. In that they attempt to answer the question "What is valuable in life?", theories of value are theories of the meaning of life: great philosophers such as Socrates, Plato, Aristotle, Descartes, Spinoza, and many others had clear views about what sort of life was best (and hence most meaningful). Meaning of life - Atheist views. Atheism in the strictest sense means the belief that no god or supernatural overbeing (of any type or number) exists ...

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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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This section aims at giving a first guide to the realm of ordered sets. It addresses readers who have basic knowledge of set theory and arithmetics and who know what a binary relation is, but who are not familiar with order theoretic considerations so far. Order theory - Partially ordered sets. As already hinted at above, orders are special binary relations. Hence consider some set P and a relation ≤ on P. Then ≤ is a partial order if it is reflexive, antisymmetric, and transitive, ...

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Order theory, Order theory - Background and motivation, Order theory - Introduction to the basic definitions, Order theory - Partially ordered sets, Order theory - Visualizing orders, Order theory - Special elements within an order, Order theory - Duality, Order theory - Constructing new orders, Order theory - Functions between orders, Order theory - Special types of orders, Order theory - Subsets of ordered sets, Order theory - Related mathematical areas, Order theory - Universal algebra, Order theory - Topology, Order theory - Category theory, Order theory - History, Order theory - Literature

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Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in gener ...

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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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Field of sets - Sigma algebras and measure spaces. If an algebra over a set is closed under countable intersections and countable unions, it is called a sigma algebra and the corresponding field of sets is called a measureable space. The complexes of a measurable space are called measureable sets. A measure space is a triple where is a measurable space and μ is a measure defined on it. If μ is in fact a probability measure w ...

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Field of sets, Field of sets - Fields of sets in the representation theory of Boolean algebras, Field of sets - Stone representation, Field of sets - Separative and compact fields of sets: towards Stone duality, Field of sets - Fields of sets with additional structure, Field of sets - Sigma algebras and measure spaces, Field of sets - Topological fields of sets, Field of sets - Preorder fields, Field of sets - Complex algebras and fields of sets on relational structures

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A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. Spelled out, this means that the following hold: Closure of F under + and *  For all a, b belonging to F, both a + b and a * b belong to F (or more formally, + and * are binary operations on F). Both + and * are associative  For all a, b, c ...

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Field mathematics, Field mathematics - Introduction, Field mathematics - Definition, Field mathematics - Examples of fields, Field mathematics - Some first theorems

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Christian left - Early Christianity as anti-establishment movement. See also: Christian anarchism Early Christian anti-materialism Opposition to Rome Christian left - Movements. A number of movements of the past had similarities to today's Christian Left: Peace churches Heretical movements such as the Cathars Waldenses Lollard, John Wycliffe Role of Christians in the Peasants' Revo ...

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Christian left, Christian left - History, Christian left - Early Christianity as anti-establishment movement, Christian left - Movements, Christian left - Early antagonism between the Left and Christianity, Christian left - Alliance of the Left and Christianity, Christian left - Christian Left and campaigns for peace and human rights, Christian left - Christian Left in the United States, Christian left - Liberation Theology, Christian left - Christian Left and Homosexuality, Christian left - The Consistent Life Ethic, Christian left - Notable Christian leftists, Christian left - Australia, Christian left - Canada, Christian left - Colombia, Christian left - El Salvador, Christian left - Germany, Christian left - Italy, Christian left - Netherlands, Christian left - Nicaragua, Christian left - New Zealand, Christian left - Russia, Christian left - Slovenia, Christian left - Switzerland, Christian left - United States, Christian left - United Kingdom, Christian left - Venezuela, Christian left - Parties of the Christian left

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Most empirical research and statistical data are oriented and tailored toward Caucasian audiences, and some studies have been designed to exclude racially diverse populations. Nonetheless, no race or socioeconomic group should be considered impervious to eating disorders. Body image - Native Americans. Native Americans have been impacted by body dissatisfaction from the influence of the media. For decades Native Americans have felt the effects of colonialization in their communities. Colonialization attemp ...

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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, Body image - Native Americans, Body image - Work cited

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Meta-ideology is the study of the structure, form, and manifestation of ideologies. Meta-ideology posits that ideology is a coherent system of ideas, relying upon a few basic assumptions about reality that may or may not have any factual basis, but are subjective choices that serve as the seed around which further thought grows. According to this perspective, ideologies are neither right nor wrong, but only a relativistic intellectual strategy for categorizing the world. The works of George Walford and Harold Walsby, done under the heading of systematic ideology, are attempts to explore t ...

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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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In the particular case of groups, congruence relations can be described in elementary terms as follows: If G is a group (with identity element e) and ~ is a binary relation on G, then ~ is a congruence whenever: Given any element a of G, a ~ a; Given any elements a and b of G, if a ~ b, then b ~ a; Given any elements a, b, and c of G, if a ~ b and b ~ c, then ...

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Congruence relation, 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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Complete lattice - Free complete semilattices. As usual, the construction of free objects depends on the chosen class of morphisms. Let us first consider functions that preserve all joins (i.e. lower adjoints of Galois connections), since this case is simpler than the situation for complete homomorphisms. Using the aforementioned terminology, this could be called a free complete join-semilattice. Using the standard definition from universal algebra, a free complete lattice over a generating set S ...

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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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Idealism names a number of philosophical positions with quite different tendencies and implications. Idealism - Idealism in the East. Several schools of Buddhism can be accurately characterized as idealist. Some of these are called "Consciousness-only" schools. Idealism - Idealism in the West. In his chief work Truth, Antiphon wrote: "Time is a thought or a measure, not a substance." This presents time as an ideational, internal, mental operation, rather t ...

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Idealism, Idealism - History, Idealism - Idealism in the East, Idealism - Idealism in the West, Idealism - Critique of Idealism, Idealism - Idealism in religious thought, Idealism - Other uses

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Given two variables X and Y, one constructs the polynomial ring R[X], and then, on top of it, the ring (R[X])[Y]. This ring is considered the polynomial ring in the two variables R[X,Y]. For example, the polynomial P(X,Y) = X2Y2 + 4XY2 + 5X3 â ...

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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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All polynomials have an expanded form, in which the distributive law has been used to remove all parentheses. (Some polynomials also have a factored form, in which parentheses appear.) In expanded form, a term of a polynomial is a part of the polynomial that includes only the operation of multiplication (where whole number powers are viewed as repeated multiplication). Every polynomial in expanded form is a sum of terms ...

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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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In 1894 he became the second World Chess Champion by defeating Steinitz with 10 wins, 4 draws and 5 losses. He maintained this title for 27 years, the longest unbroken tenure of any officially recognized World Champion of chess. His great tournament wins include London (1899), St Petersburg (1896 and 1914), New York (1924). In 1921, he lost the title to Capablanca. He had already offered to resign to him a year before, but Capablanca wanted to beat Lasker in a match. In 1933, the Jewish Lasker and his wife Martha Kohn had to leave Germany because of the Nazis. They went to England, and, after a subsequent ...

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Emanuel Lasker, Emanuel Lasker - Chess champion, Emanuel Lasker - Mathematician, Emanuel Lasker - Other facets of his life, Emanuel Lasker - Books, Emanuel Lasker - Quotations

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Vladimir Lenin continued and extended the use of the word "decadence" in his theory of imperialism to refer to economic matters underlying political manifestations. According to Lenin, capitalism had reached its highest stage and could no longer provide for general development of society. He expected reduced vigor in economic activity and a growth in unhealthy economic phenomenon, because society was ripe for socialist revolution in the West. Politically, World War I proved the decadent nature of the advanced capitalist countries to Lenin, that capitalism had reached the stage where it would destroy its own prio ...

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Decadence, Decadence - Leninist use, Decadence - Decadent artists

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