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Krull–Schmidt Theorem
In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable subgroups. Definitions We say that a group ''G'' satisfies the ascending chain condition (ACC) on subgroups if every sequence of subgroups of ''G'': :1 = G_0 \le G_1 \le G_2 \le \cdots\, is eventually constant, i.e., there exists ''N'' such that ''G''''N'' = ''G''''N''+1 = ''G''''N''+2 = ... . We say that ''G'' satisfies the ACC on normal subgroups if every such sequence of normal subgroups of ''G'' eventually becomes constant. Likewise, one can define the descending chain condition on (normal) subgroups, by looking at all decreasing sequences of (normal) subgroups: :G = G_0 \ge G_1 \ge G_2 \ge \cdots.\, Clearly, all finite groups satisfy both ACC and DCC on subgroups. The infinite cyclic group \mathbf satisfies ACC but not DCC, since (2) > (2)2 >&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Module
In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated. However, the property is named after Emmy Noether who was the first one to discover the true importance of the property. Characterizations and properties In the presence of the axiom of choice, two other characterizations are possible: *Any nonempty set ''S'' of submodules of the module has a maximal element (with respect to set inclusion). This is known as the maximum condition. *All of the submodules of the module are finitely generated. If ''M'' is a module and ''K'' a submodule, then ''M'' is Noetherian if and only if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull–Schmidt Category
In category theory, a branch of mathematics, a Krull–Schmidt category is a generalization of categories in which the Krull–Schmidt theorem holds. They arise, for example, in the study of finite-dimensional module (mathematics), modules over an algebra over a field, algebra. Definition Let ''C'' be an additive category, or more generally an additive preadditive category#R-linear categories, -linear category for a commutative ring . We call ''C'' a Krull–Schmidt category provided that every object decomposes into a finite direct sum of objects having local endomorphism rings. Equivalently, ''C'' has split idempotents and the endomorphism ring of every object is perfect ring#semiperfect ring, semiperfect. Properties One has the analogue of the Krull–Schmidt theorem in Krull–Schmidt categories: An object is called ''indecomposable'' if it is not isomorphic to a direct sum of two nonzero objects. In a Krull–Schmidt category we have that *an object is indecomposabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goro Azumaya
was a Japanese mathematician who introduced the notion of Azumaya algebra in 1951. His advisor was Shokichi Iyanaga. At the time of his death he was an emeritus professor at Indiana University Indiana University (IU) is a system of public universities in the U.S. state of Indiana. Campuses Indiana University has two core campuses, five regional campuses, and two regional centers under the administration of IUPUI. *Indiana Universi .... References External links * Biography of Azumayaby BiRep, Bielefeld University 1920 births 20th-century Japanese mathematicians 21st-century Japanese mathematicians Algebraists Indiana University faculty 2010 deaths Nagoya University alumni Japanese expatriates in the United States {{Asia-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas W
Thomas may refer to: People * List of people with given name Thomas * Thomas (name) * Thomas (surname) * Saint Thomas (other) * Thomas Aquinas (1225–1274) Italian Dominican friar, philosopher, and Doctor of the Church * Thomas the Apostle * Thomas (bishop of the East Angles) (fl. 640s–650s), medieval Bishop of the East Angles * Thomas (Archdeacon of Barnstaple) (fl. 1203), Archdeacon of Barnstaple * Thomas, Count of Perche (1195–1217), Count of Perche * Thomas (bishop of Finland) (1248), first known Bishop of Finland * Thomas, Earl of Mar (1330–1377), 14th-century Earl, Aberdeen, Scotland Geography Places in the United States * Thomas, Illinois * Thomas, Indiana * Thomas, Oklahoma * Thomas, Oregon * Thomas, South Dakota * Thomas, Virginia * Thomas, Washington * Thomas, West Virginia * Thomas County (other) * Thomas Township (other) Elsewhere * Thomas Glacier (Greenland) Arts, entertainment, and media * ''Thomas'' (Burton novel) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject. Krull was born and went to school in Baden-Baden. He attended the Universities of Freiburg, Rostock and finally Göttingen from 1919–1921, where he earned his doctorate under Alfred Loewy. He worked as an instructor and professor at Freiburg, then spent a decade at the University of Erlangen. In 1939 Krull moved to become chair at the University of Bonn, where he remained for the rest of his life. Wolfgang Krull was a member of the Nazi Party. His 35 doctoral students include Wilfried Brauer, Karl-Otto Stöhr and Jürgen Neukirch. See also * Cohen structure theorem * Jacobson ring * Local ring * Prime ideal * Real algebraic geometry * Regular local ring * Valuation ring * Krull dimension * Krull ring * Krull topology In mathematics, a profinite group is a topological group that is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Otto Schmidt
Otto Yulyevich Shmidt, be, Ота Юльевіч Шміт, Ota Juljevič Šmit (born Otto Friedrich Julius Schmidt; – 7 September 1956), better known as Otto Schmidt, was a Soviet scientist, mathematician, astronomer, geophysicist, statesman, and academician. Biography He was born in the town of Mogilev in the Russian Empire, in what is now Belarus. His father was a descendant of German settlers in Courland, while his mother was a Latvian. In 1912-13 while in university he published a number of mathematical works on group theory which laid foundation for Krull–Schmidt theorem. In 1913, Schmidt married Vera Yanitskaia and graduated from the Saint Vladimir Imperial University of Kiev, where he worked as a privat-docent starting from 1916. In 1918 he became a member of the Russian Social Democratic Labour Party (internationallists) which was later dissolved in to the Russian Communist Party (b). After the October Revolution of 1917, he was a board member at severa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Remak Decomposition
Remak is a surname. Notable people with the surname include: * Ernst Remak (1849-1911), German neurologist, son of Robert Remak * Joachim Remak (b. 1920), German-American historian of World War I * Patricia Remak (b. 1965), former Dutch politician * Robert Remak (1815-1865), Polish/German neurologist, zoologist * Robert Remak (1888–1942), German mathematician, son of Ernst Remak See also * Moses ben Jacob Cordovero Moses ben Jacob Cordovero ( he, משה קורדובירו ''Moshe Kordovero'' ; 1522–1570) was a central figure in the historical development of Kabbalah, leader of a mystical school in 16th-century Safed, Ottoman Syria. He is known by th ..., known as ''Ramak'' * REMAK, the computer from the episode "Killer" of the British TV series '' The Avengers'' * Remake {{surname, Remak Jewish surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Abram Miller
George Abram Miller (31 July 1863 – 10 February 1951) was an early group theorist. At age 17 Miller began school-teaching to raise funds for higher education. In 1882 he entered Franklin and Marshall Academy, and progressed to Muhlenberg College in 1884. He received his B.A. in 1887 and M.A. in 1890. While a graduate student, Miller was Principal of schools in Greeley, Kansas and then professor of mathematics as Eureka College in Eureka, Illinois. He corresponded with Cumberland University in Lebanon, Tennessee for his Ph.D. in 1892. He then joined Frank Nelson Cole at University of Michigan and began to study groups. In 1895 he went to Europe where he heard Sophus Lie lecture at Leipzig and Camille Jordan at Paris. In 1897 he went to Cornell University as an assistant professor, and in 1901 to Stanford University as associate professor. In 1906 he went to University of Illinois where he taught until retirement in 1931. Henry Roy Brahana (1957) Miller helped in the enumerat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Wedderburn
Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras. He also worked on group theory and matrix algebra. His younger brother was the lawyer Ernest Wedderburn. Life Joseph Wedderburn was the tenth of fourteen children of Alexander Wedderburn of Pearsie, a physician, and Anne Ogilvie. He was educated at Forfar Academy then in 1895 his parents sent Joseph and his younger brother Ernest to live in Edinburgh with their paternal uncle, J R Maclagan Wedderburn, allowing them to attend George Watson's College. This house was at 3 Glencairn Crescent in the West End of the city. In 1898 Joseph entered the University of Edinburgh. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indecomposable Module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. Jacobson (2009), p. 111. Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple means "no proper submodule" N < M, while indecomposable "not expressible as ". A direct sum of indecomposables is called completely decomposable; this is weaker than being semisimple, which is a direct sum of simple modules. A direct sum decomposition of a module into indecomposable modules is called an indecomposa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Sum Of Modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers). The construction may also be extended to cover Banach spaces and Hilbert spaces. See the article decomposition of a module for a way to write a module as a direct sum of submodules. Construction for vector spaces and abelian groups We give the construction first in these two cases, under the assumption that we have only two objects. Then we generalize to an arbitrary family of arbitrary modules. The key elements of the general construction are more clearly identified by consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
