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Radical Of An Ideal
Radical (from Latin: ', root) may refer to: Politics and ideology Politics * Classical radicalism, the Radical Movement that began in late 18th century Britain and spread to continental Europe and Latin America in the 19th century * Radical politics, the political intent of fundamental societal change * Radical Party (other), several political parties *Radicals (UK), a British and Irish grouping in the early to mid-19th century * Radicalization *Politicians from the Radical Civic Union Ideologies * Radical chic, a term coined by Tom Wolfe to describe the pretentious adoption of radical causes * Radical feminism, a perspective within feminism that focuses on patriarchy * Radical Islam, or Islamic extremism * Radical Christianity * Radical veganism, a radical interpretation of veganism, usually combined with anarchism * Radical Reformation, an Anabaptist movement concurrent with the Protestant Reformation Science and mathematics Science * Radical (chemistry), an atom, molec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical Radicalism
Radicalism (from French ) was a political movement representing the leftward flank of liberalism between the late 18th and early 20th century. Certain aspects of the movement were precursors to a wide variety of modern-day movements, ranging from ''laissez-faire'' to social liberalism, social democracy, civil libertarianism, and modern progressivism. This ideology is commonly referred to as "radicalism" but is sometimes referred to as radical liberalism, or classical radicalism, to distinguish it from radical politics. Its earliest beginnings are to be found during the English Civil War with the Levellers and later the Radical Whigs. During the 19th century in the United Kingdom, continental Europe and Latin America, the term ''radical'' came to denote a progressive liberal ideology inspired by the French Revolution. Radicalism grew prominent during the 1830s in the United Kingdom with the Chartists and in Belgium with the Revolution of 1830, then across Europe in the 1840s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of An Algebraic Group
The radical of an algebraic group is the identity component of its maximal normal solvable subgroup. For example, the radical of the general linear group \operatorname_n(K) (for a field ''K'') is the subgroup consisting of scalar matrices, i.e. matrices (a_) with a_ = \dots = a_ and a_=0 for i \ne j. An algebraic group is called semisimple if its radical is trivial, i.e., consists of the identity element only. The group \operatorname_n(K) is semi-simple, for example. The subgroup of unipotent elements in the radical is called the unipotent radical, it serves to define reductive groups. See also * Reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation that has a finite kernel and is a ... * Unipotent group References "Radical of a group" Encyclopaedia of Mathematics Algebraic groups {{grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semitic Root
The roots of verbs and most nouns in the Semitic languages are characterized as a sequence of consonants or " radicals" (hence the term consonantal root). Such abstract consonantal roots are used in the formation of actual words by adding the vowels and non-root consonants (or "transfixes"), which go with a particular morphological category around the root consonants, in an appropriate way, generally following specific patterns. It is a peculiarity of Semitic linguistics that many of these consonantal roots are triliterals, meaning that they consist of three letters (although there are a number of quadriliterals, and in some languages also biliterals). Such roots are also common in other Afroasiatic languages. While Berber mostly has triconsonantal roots, Chadic, Omotic, and Cushitic have mostly biconsonantal roots; and Egyptian shows a mix of biconsonantal and triconsonantal roots. Triconsonantal roots A triliteral or triconsonantal root (; , ';, '; , ') is a root containing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Consonant
A pharyngeal consonant is a consonant that is articulated primarily in the pharynx. Some phoneticians distinguish upper pharyngeal consonants, or "high" pharyngeals, pronounced by retracting the root of the tongue in the mid to upper pharynx, from (ary)epiglottal consonants, or "low" pharyngeals, which are articulated with the aryepiglottic folds against the epiglottis at the entrance of the larynx, as well as from epiglotto-pharyngeal consonants, with both movements being combined. Stops and trills can be reliably produced only at the epiglottis, and fricatives can be reliably produced only in the upper pharynx. When they are treated as distinct places of articulation, the term ''radical consonant'' may be used as a cover term, or the term '' guttural consonants'' may be used instead. Pharyngeal consonants can trigger effects on neighboring vowels. Instead of uvulars, which nearly always trigger retraction, pharyngeals tend to trigger lowering. For example, in Moroccan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical (Chinese Characters)
A radical (), or indexing component, is a visually prominent Chinese character components, component of a Chinese character under which the character is traditionally listed in a Chinese dictionary. The radical for a character is typically a semantic component, but it can also be another structural component or an artificially extracted portion of the character. In some cases, the original semantic or phonological connection has become obscure, owing to changes in the meaning or pronunciation of the character over time. The use of the English term ''radical'' is based on an analogy between the structure of Chinese characters and the inflection of words in European languages. Radicals are also sometimes called ''classifiers'', but this name is more commonly applied to the grammatical Chinese classifier, measure words in Chinese. History In the earliest Chinese dictionaries, such as the ''Erya'' (3rd centuryBC), characters were grouped together in broad semantic categories. Be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Root (linguistics)
A root (also known as a root word or radical) is the core of a word that is irreducible into more meaningful elements. In morphology, a root is a morphologically simple unit which can be left bare or to which a prefix or a suffix can attach. The root word is the primary lexical unit of a word, and of a word family (this root is then called the base word), which carries aspects of semantic content and cannot be reduced into smaller constituents. Content words in nearly all languages contain, and may consist only of, root morphemes. However, sometimes the term "root" is also used to describe the word without its inflectional endings, but with its lexical endings in place. For example, ''chatters'' has the inflectional root or lemma ''chatter'', but the lexical root ''chat''. Inflectional roots are often called stems. A root, or a root morpheme, in the stricter sense, is a mono-morphemic stem. The traditional definition allows roots to be either free morphemes or bound ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linear in each argument separately: * and * and The dot product on \R^n is an example of a bilinear form which is also an inner product. An example of a bilinear form that is not an inner product would be the four-vector product. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When is the field of complex numbers , one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument. Coordinate representation Let be an - dimensional vector space with basis . The matrix ''A'', defined by is called the ''matrix of the bilinear form'' on the basis . If the matrix represents a ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilradical Of A Lie Algebra
In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical \mathfrak(\mathfrak g) of a finite-dimensional Lie algebra \mathfrak is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the radical \mathfrak(\mathfrak) of the Lie algebra \mathfrak. The quotient of a Lie algebra by its nilradical is a reductive Lie algebra \mathfrak^. However, the corresponding short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ... : 0 \to \mathfrak(\mathfrak g)\to \mathfrak g\to \mathfrak^\to 0 does not split in general (i.e., there isn't always a ''subalgebra'' complementary to \mathfrak(\mathfrak g) in \mathfrak). This is in contrast to the Levi decomposition: th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of A Lie Algebra
In the mathematical field of Lie theory, the radical of a Lie algebra \mathfrak is the largest solvable ideal of \mathfrak.. The radical, denoted by (\mathfrak), fits into the exact sequence :0 \to (\mathfrak) \to \mathfrak g \to \mathfrak/(\mathfrak) \to 0. where \mathfrak/(\mathfrak) is semisimple. When the ground field has characteristic zero and \mathfrak g has finite dimension, Levi's theorem states that this exact sequence splits; i.e., there exists a (necessarily semisimple) subalgebra of \mathfrak g that is isomorphic to the semisimple quotient \mathfrak/(\mathfrak) via the restriction of the quotient map \mathfrak g \to \mathfrak/(\mathfrak). A similar notion is a Borel subalgebra, which is a (not necessarily unique) maximal solvable subalgebra. Definition Let k be a field and let \mathfrak be a finite-dimensional Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of An Integer
In number theory, the radical of a positive integer ''n'' is defined as the product of the distinct prime numbers dividing ''n''. Each prime factor of ''n'' occurs exactly once as a factor of this product: \displaystyle\mathrm(n)=\prod_p The radical plays a central role in the statement of the abc conjecture. Examples Radical numbers for the first few positive integers are : 1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, ... . For example, 504 = 2^3 \cdot 3^2 \cdot 7 and therefore \operatorname(504) = 2 \cdot 3 \cdot 7 = 42 Properties The function \mathrm is multiplicative (but not completely multiplicative). The radical of any integer n is the largest square-free divisor of n and so also described as the square-free kernel of n. There is no known polynomial-time algorithm for computing the square-free part of an integer. The definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of A Module
In mathematics, in the theory of modules, the radical of a module is a component in the theory of structure and classification. It is a generalization of the Jacobson radical for rings. In many ways, it is the dual notion to that of the socle soc(''M'') of ''M''. Definition Let ''R'' be a ring and ''M'' a left ''R''- module. A submodule ''N'' of ''M'' is called maximal or cosimple if the quotient ''M''/''N'' is a simple module. The radical of the module ''M'' is the intersection of all maximal submodules of ''M'', :\mathrm(M) = \bigcap\, \ Equivalently, :\mathrm(M) = \sum\, \ These definitions have direct dual analogues for soc(''M''). Properties * In addition to the fact rad(''M'') is the sum of superfluous submodules, in a Noetherian module rad(''M'') itself is a superfluous submodule. In fact, if ''M'' is finitely generated over a ring, then rad(''M'') itself is a superfluous submodule. This is because any proper submodule of ''M'' is contained in a maximal submodu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilradical Of A Ring
In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements: :\mathfrak_R = \lbrace f \in R \mid f^m=0 \text m\in\mathbb_\rbrace. It is thus the radical of the zero ideal. If the nilradical is the zero ideal, the ring is called a reduced ring. The nilradical of a commutative ring is the intersection of all prime ideals. In the non-commutative ring case the same definition does not always work. This has resulted in several radicals generalizing the commutative case in distinct ways; see the article Radical of a ring for more on this. The nilradical of a Lie algebra is similarly defined for Lie algebras. Commutative rings The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent (by the binomial formula), and the product of any element with a nilpotent element is nilpotent (by commu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
