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Kurosh Conjecture
In mathematics, the Kurosh problem is one general problem, and several more special questions, in ring theory. The general problem is known to have a negative solution, since one of the special cases has been shown to have counterexamples. These matters were brought up by Aleksandr Gennadievich Kurosh as analogues of the Burnside problem in group theory. Kurosh asked whether there can be a finitely-generated infinite-dimensional algebraic algebra (the problem being to show this cannot happen). A special case is whether or not every nil algebra is locally nilpotent. For PI-algebras the Kurosh problem has a positive solution. Golod showed a counterexample to that case, as an application of the Golod–Shafarevich theorem. The Kurosh problem on group algebras concerns the augmentation ideal ''I''. If ''I'' is a nil ideal, is the group algebra locally nilpotent? There is an important problem which is often referred as the Kurosh's problem on division rings. The problem asks wheth ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golod–Shafarevich Theorem
In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite. The inequality Let ''A'' = ''K''⟨''x''1, ..., ''x''''n''⟩ be the free algebra over a field ''K'' in ''n'' = ''d'' + 1 non-commuting variables ''x''''i''. Let ''J'' be the 2-sided ideal of ''A'' generated by homogeneous elements ''f''''j'' of ''A'' of degree ''d''''j'' with :2 ≤ ''d''1 ≤ ''d''2 ≤ ... where ''d''''j'' tends to infinity. Let ''r''''i'' be the number of ''d''''j'' equal to ''i''. Let ''B''=''A''/''J'', a graded algebra. Let ''b''''j'' = dim ''B''''j''. The ''fundamental inequality'' of Golod and Shafarevich states that :: b_j\ge nb_ -\sum_^ b_ r_i. As a consequence: * ''B'' is infinite-dimensional if ''r''''i'' ≤ ''d''2/4 for all ''i'' Applications This result has important applicatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Efim Zelmanov
Efim Isaakovich Zelmanov (russian: Ефи́м Исаа́кович Зе́льманов; born 7 September 1955 in Khabarovsk) is a Russian-American mathematician, known for his work on combinatorial problems in nonassociative algebra and group theory, including his solution of the restricted Burnside problem. He was awarded a Fields Medal at the International Congress of Mathematicians in Zürich in 1994. Zelmanov was born into a Jewish family in Khabarovsk, Soviet Union (now in Russia). He entered Novosibirsk State University in 1972, when he was 17 years old. He obtained a doctoral degree at Novosibirsk State University in 1980, and a higher degree at Leningrad State University in 1985. He had a position in Novosibirsk until 1987, when he left the Soviet Union.In 1990 he moved to the United States, becoming a professor at the University of Wisconsin–Madison. He was at the University of Chicago in 1994/5, then at Yale University. In 2011, he became a professor at the University ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edward W
Edward is an English given name. It is derived from the Anglo-Saxon name ''Ēadweard'', composed of the elements '' ēad'' "wealth, fortune; prosperous" and '' weard'' "guardian, protector”. History The name Edward was very popular in Anglo-Saxon England, but the rule of the Norman and Plantagenet dynasties had effectively ended its use amongst the upper classes. The popularity of the name was revived when Henry III named his firstborn son, the future Edward I, as part of his efforts to promote a cult around Edward the Confessor, for whom Henry had a deep admiration. Variant forms The name has been adopted in the Iberian peninsula since the 15th century, due to Edward, King of Portugal, whose mother was English. The Spanish/Portuguese forms of the name are Eduardo and Duarte. Other variant forms include French Édouard, Italian Edoardo and Odoardo, German, Dutch, Czech and Romanian Eduard and Scandinavian Edvard. Short forms include Ed, Eddy, Eddie, Ted, Teddy and Ned. Pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Center (ring Theory)
In algebra, the center of a ring ''R'' is the subring consisting of the elements ''x'' such that ''xy = yx'' for all elements ''y'' in ''R''. It is a commutative ring and is denoted as Z(R); "Z" stands for the German word ''Zentrum'', meaning "center". If ''R'' is a ring, then ''R'' is an associative algebra over its center. Conversely, if ''R'' is an associative algebra over a commutative subring ''S'', then ''S'' is a subring of the center of ''R'', and if ''S'' happens to be the center of ''R'', then the algebra ''R'' is called a central algebra. Examples *The center of a commutative ring ''R'' is ''R'' itself. *The center of a skew-field is a field. *The center of the (full) matrix ring with entries in a commutative ring ''R'' consists of ''R''-scalar multiples of the identity matrix. *Let ''F'' be a field extension of a field ''k'', and ''R'' an algebra over ''k''. Then Z\left(R \otimes_k F\right) = Z(R) \otimes_k F. *The center of the universal enveloping algebra of a Lie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Division Ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element usually denoted , such that . So, (right) ''division'' may be defined as , but this notation is avoided, as one may have . A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields". In some languages, such as French, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field). All division rings are simple. That is, they have no two-sided ideal besi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nil Ideal
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent., p. 194 The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nil elements does not always form an ideal for noncommutative rings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture. Commutative rings In commutative rings, the nil ideals are better understood than in noncommutative rings, primarily because in commutative rings, products involving nilpotent elements and sums of nilpotent elements are both nilpotent. This is because if ''a'' and ''b'' are nilpotent elements of ''R'' with ''a''n=0 and ''b''m=0, and r is any element of R, then (''a''·''r'')n = ''a''n·''r''n = 0, and by the binomial theorem, (''a''+''b'')m+n=0. Therefore, the set of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augmentation Ideal
In algebra, an augmentation ideal is an ideal that can be defined in any group ring. If ''G'' is a group and ''R'' a commutative ring, there is a ring homomorphism \varepsilon, called the augmentation map, from the group ring R /math> to R, defined by taking a (finiteWhen constructing , we restrict to only finite (formal) sums) sum \sum r_i g_i to \sum r_i. (Here r_i\in R and g_i\in G.) In less formal terms, \varepsilon(g)=1_R for any element g\in G, \varepsilon(r) = r for any element r\in R, and \varepsilon is then extended to a homomorphism of ''R''-modules in the obvious way. The augmentation ideal is the kernel of \varepsilon and is therefore a two-sided ideal in ''R'' 'G'' is generated by the differences g - g' of group elements. Equivalently, it is also generated by \, which is a basis as a free ''R''-module. For ''R'' and ''G'' as above, the group ring ''R'' 'G''is an example of an ''augmented'' ''R''-algebra. Such an algebra comes equipped with a ring homomorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring. If the ring is commutative then the group ring is also referred to as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case, it is thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Definition Let ''G'' be a group, written multiplicatively, and let ''R'' be a ring. The group ring of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evgeny Golod
Evgenii Solomonovich Golod (russian: Евгений Соломонович Голод, 21 October 1935 – 5 July 2018) was a Russian mathematician who proved the Golod–Shafarevich theorem on class field tower A tower is a tall structure, taller than it is wide, often by a significant factor. Towers are distinguished from masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting structures. Towers are specifi ...s. As an application, he gave a negative solution to the Kurosh–Levitzky problem on the nilpotency of finitely generated nil algebras, and so to a weak form of Burnside's problem. Golod was a student of Igor Shafarevich. As of 2015, Golod had 39 academic descendants, most of them through his student Luchezar L. Avramov. Selected publications * * References {{DEFAULTSORT:Golod, Evgeny S. 1935 births 2018 deaths Russian mathematicians Scientists from Moscow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Theory
In algebra, ring theory is the study of rings— algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological algebra, homological properties and Polynomial identity ring, polynomial identities. Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of ''commutative algebra'', a major area of modern mathematics. Because these three fields (algebraic geometry, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PI-algebra
In ring theory, a branch of mathematics, a ring ''R'' is a polynomial identity ring if there is, for some ''N'' > 0, an element ''P'' ≠ 0 of the free algebra, Z, over the ring of integers in ''N'' variables ''X''1, ''X''2, ..., ''X''''N'' such that :P(r_1, r_2, \ldots, r_N) = 0 for all ''N''-tuples ''r''1, ''r''2, ..., ''r''''N'' taken from ''R''. Strictly the ''X''''i'' here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring ''S'' may be used, and gives the concept of PI-algebra. If the degree of the polynomial ''P'' is defined in the usual way, the polynomial ''P'' is called monic if at least one of its terms of highest degree has coefficient equal to 1. Every commutative ring is a PI-ring, satisfying the polynomial identity ''XY'' − ''YX'' = 0. Therefore, P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
