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Artinian (other)
{{Disambiguation, math, surname ...
Artinian may refer to: Mathematics *Objects named for Austrian mathematician Emil Artin (1898–1962) **Artinian ideal, an ideal ''I'' in ''R'' for which the Krull dimension of the quotient ring ''R/I'' is 0 **Artinian ring, a ring which satisfies the descending chain condition on (one-sided) ideals **Artinian module, a module which satisfies the descending chain condition on submodules **Artinian group, a group which satisfies the descending chain condition on subgroups People *Araz Artinian, Armenian-Canadian filmmaker and photographer *Artine Artinian (1907–2005), French literature scholar See also *Descending chain condition *List of things named after Emil Artin {{Short description, none These are things named after Emil Artin, a mathematician. * Ankeny–Artin–Chowla congruence * Artin algebra * Artin billiards * Artin braid group * Artin character * Artin conductor * Artin's conjecture for conjectu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing largely to class field theory and a new construction of L-functions. He also contributed to the pure theories of rings, groups and fields. Along with Emmy Noether, he is considered the founder of modern abstract algebra. Early life and education Parents Emil Artin was born in Vienna to parents Emma Maria, née Laura (stage name Clarus), a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. His Armenian last name was Artinian which was shortened to Artin. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible that he and Emma had ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Ideal
In abstract algebra, an Artinian ideal, named after Emil Artin, is encountered in ring theory, in particular, with polynomial rings. Given a polynomial ring ''R'' = ''k'' 'X''1, ... ''X''''n''where ''k'' is some field, an Artinian ideal is an ideal ''I'' in ''R'' for which the Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally t ... of the quotient ring ''R''/''I'' is 0. Also, less precisely, one can think of an Artinian ideal as one that has at least each indeterminate in ''R'' raised to a power greater than 0 as a generator. If an ideal is not Artinian, one can take the Artinian closure of it as follows. First, take the least common multiple of the generators of the ideal. Second, add to the generating set of the ideal each indeterminate of the LCM ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Artin–Wedderburn theorem charact ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Module
In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named for Emil Artin. In the presence of the axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ..., the descending chain condition becomes equivalent to the minimum condition, and so that may be used in the definition instead. Like Noetherian modules, Artinian modules enjoy the following heredity property: * If ''M'' is an Artinian ''R''-module, then so is any submodule and any quotient module, quotient of ''M''. The converse (logic), co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Group
In mathematics, specifically group theory, a subgroup series of a group G is a chain of subgroups: :1 = A_0 \leq A_1 \leq \cdots \leq A_n = G where 1 is the trivial subgroup. Subgroup series can simplify the study of a group to the study of simpler subgroups and their relations, and several subgroup series can be invariantly defined and are important invariants of groups. A subgroup series is used in the subgroup method. Subgroup series are a special example of the use of filtrations in abstract algebra. Definition Normal series, subnormal series A subnormal series (also normal series, normal tower, subinvariant series, or just series) of a group ''G'' is a sequence of subgroups, each a normal subgroup of the next one. In a standard notation :1 = A_0\triangleleft A_1\triangleleft \cdots \triangleleft A_n = G. There is no requirement made that ''A''''i'' be a normal subgroup of ''G'', only a normal subgroup of ''A''''i'' +1. The quotient groups ''A''''i'' +1/''A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Araz Artinian
Araz Artinian is a Canadian filmmaker and documentarian of Armenian descent.Araz Artinian's "The Genocide" - The Armenian reporter She was born Montreal, Quebec. In 1999, she started working with Atom Egoyan as Head Researcher for his feature film Ararat (film), Ararat which was premiered at the 2002 Cannes Film Festival. ''The Genocide in Me'' is the latest documentary she wrote, filmed, and directed. References External linksThe Genocide in Me - Artinian's documentary website Araz Artinian at ArmeniaPedia.org {{DEFAULTSORT:Artinian, Araz Year of birth missing (living people) Living people Canadian people of Armenian descent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artine Artinian
Artine Artinian (December 8, 1907 – November 19, 2005) was a distinguished French literature scholar of Armenian descent, notable for his valuable collection of French literary manuscripts and artwork. He was immortalized as a fictional character by his Bard colleague Mary McCarthy in the novel ''The Groves of Academe'' (1952) and by his friend Gore Vidal in the play '' The Best Man'' (1960). Background Artine Artinian was born in Pazardzhik, Bulgaria to Armenian parents. In 1920, his family came to the United States, settling in Attleboro, Massachusetts. There, Artine worked as a shoeshine boy, learning English from listening to conversations as he worked. He was able to attend Bowdoin College (1931) with support from his customers, and in later years, he returned the favor by establishing a scholarship fund for needy students there. He received a diploma from the Université de Paris in 1932, an A.M. from Harvard the following year, and a Ph.D. from Columbia in 1941. His disse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descending Chain Condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler. Definition A partially ordered set (poset) ''P'' is said to satisfy the ascending chain condition (ACC) if no infinite strictly ascending sequence :a_1 < a_2 < a_3 < \cdots of elements of ''P'' exists. Equivalently,Proof: first, a strictly increasing sequence cannot stabilize, obviously. Conversely, suppose there is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |