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Krull (other)
Krull is a surname originating from Prussian nobility. People *Alexander Krull (born 1970), German singer * Annie Krull (1876–1947), German operatic soprano *Germaine Krull (1897–1985), photographer *Hasso Krull (born 1964), Estonian poet, literary and cultural critic and translator * Jake Krull (1938–2016), American politician *Kathleen Krull (born 1952), American author of children's books * Lucas Krull (born 1998), American football player *Reinhard Krull (born 1954), West German field hockey player *Suzanne Krull (born 1966), American actress * Wolfgang Krull, German mathematician, who was responsible for the development of numerous mathematical concepts: ** Krull dimension ** Krull's principal ideal theorem ** Krull's theorem **Krull–Akizuki theorem ** Krull–Schmidt theorem ** Krull topology, an example of the profinite group ** Krull's intersection, a theorem within algebraic ring theory that describes the behaviors of certain local rings In fiction and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Krull
Alexander Krull (born 31 July 1970) is a German musician who is the lead vocalist for metal band Atrocity, as well as backing vocalist in the band Leaves' Eyes. Biography Krull was born in Ludwigsburg, where he formed Atrocity in 1985. He has also worked as a music producer for Elis, Leaves' Eyes, and Erben der Schöpfung. He uses the Mastersound Studio for his recording work with the bands. Personal life Krull was married to Liv Kristine of the Norwegian gothic metal band Theatre of Tragedy on 3 July 2003. Kristine gave birth to their first and only son the same year in December. The couple split in January 2016. Krull's sister Yasmin has performed as guest singer with Atrocity on two projects and works as a |
Krull's Principal Ideal Theorem
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is sometimes referred to by its German name, ''Krulls Hauptidealsatz'' ('' Satz'' meaning "proposition" or "theorem"). Precisely, if ''R'' is a Noetherian ring and ''I'' is a principal, proper ideal of ''R'', then each minimal prime ideal over ''I'' has height at most one. This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem. This says that if ''R'' is a Noetherian ring and ''I'' is a proper ideal generated by ''n'' elements of ''R'', then each minimal prime over ''I'' has height at most ''n''. The converse is also true: if a prime ideal has height ''n'', then it is a minimal prime ideal over an ideal generated by ''n'' elements. The principal ideal theorem and the generalization, the height theorem, both follow from the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull (video Game)
''Krull'' is an Atari 2600 video game based on the 1983 science fantasy film '' Krull'' and published in 1983 by Atari, Inc. It was written by Dave Staugas who later ported ''Millipede'' to the 2600. Gottlieb Gottlieb (formerly D. Gottlieb & Co.) was an American arcade game corporation based in Chicago, Illinois. History The main office and plant was located at 1140-50 N. Kostner Avenue until the early 1970s when a new modern plant and office was lo ... manufactured an arcade shooter of the same name in the same year, but it is unrelated to the Atari 2600 cartridge other than the ''Krull'' license. Gameplay The game generally follows the plot of the movie, and takes place on four separate screens. The first level begins with the player, as Colwyn, at his wedding to Lyssa, which is interrupted by the extraterrestrial Slayers. The game continues to generate new Slayers for the player to fight until he is overwhelmed and Lyssa is abducted to the Black Fortress.http://www.atari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull (film)
''Krull'' is a 1983 science fantasy swashbuckler filmNathan, Ian (10 October 2015)"Krull review" ''Empire''. Bauer Media Group. Retrieved 28 August 2017. directed by Peter Yates and written by Stanford Sherman. It follows Prince Colwyn and a fellowship of companions who set out to rescue his bride, Princess Lyssa, from a fortress of alien invaders who have arrived on their home planet. The film stars an ensemble cast, including Kenneth Marshall as Prince Colwyn, Lysette Anthony as Princess Lyssa, Trevor Martin as the voice of the Beast, Freddie Jones as Ynyr, Bernard Bresslaw as Rell the Cyclops, David Battley as Ergo the Magnificent, Tony Church and Bernard Archard as kings and the fathers of Colwyn and Lyssa, Alun Armstrong as the leader of a group of bandits that include early screen roles for actors Liam Neeson and Robbie Coltrane, John Welsh as The Emerald Seer, Graham McGrath as Titch, and Francesca Annis as The Widow of the Web. Development on the film began in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Ring
In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name ''Stellenringe''. The English term ''local ring'' is due to Zariski. Definition and first consequences A ring ''R'' is a local ring if it has any one of the following equivalent properties: * ''R'' has a unique maximal left ideal. * ''R'' has a unique maximal right ideal. * 1 ≠ 0 and the sum of any two non-units in ''R'' is a non-unit. * 1 ≠ 0 and if ''x'' is any element of ''R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Profinite Group
In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists d\in\N such that every group in the system can be generated by d elements. Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems. To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the profini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Krull–Akizuki Theorem
In algebra, the Krull–Akizuki theorem states the following: let ''A'' be a one-dimensional reduced noetherian ring, ''K'' its total ring of fractions. If ''B'' is a subring of a finite extension ''L'' of ''K'' containing ''A'' then ''B'' is a one-dimensional noetherian ring. Furthermore, for every nonzero ideal ''I'' of ''B'', B/I is finite over ''A''. Note that the theorem does not say that ''B'' is finite over ''A''. The theorem does not extend to higher dimension. One important consequence of the theorem is that the integral closure of a Dedekind domain ''A'' in a finite extension of the field of fractions of ''A'' is again a Dedekind domain. This consequence does generalize to a higher dimension: the Mori–Nagata theorem states that the integral closure of a noetherian domain is a Krull domain In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull's Theorem
In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring has at least one maximal ideal. The theorem was proved in 1929 by Krull, who used transfinite induction. The theorem admits a simple proof using Zorn's lemma, and in fact is equivalent to Zorn's lemma, which in turn is equivalent to the axiom of choice. Variants * For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold. * For pseudo-rings, the theorem holds for regular ideals. * A slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows: :::Let ''R'' be a ring, and let ''I'' be a proper ideal of ''R''. Then there is a maximal ideal of ''R'' containing ''I''. :This result implies the original theorem, by taking ''I'' to be the zero ideal (0). Conversely, applying the original theorem to ''R''/''I'' leads to this result. :To prove the stronger result directly, consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal ''I'' in a polynomial ring ''R'' is the Krull dimension of ''R''/''I''. A field ''k'' has Krull dimension 0; more generally, ''k'' 'x''1, ..., ''x''''n''has Krull dimension ''n''. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent. There are several other ways that have been used to define the dimension of a ring. Most of them coinci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annie Krull
Anna Maria Krull (12 January 1876 – 14 June 1947) was a German operatic soprano. She is most remembered today for having created the title role in Richard Strauss' opera '' Elektra''. Biography Annie Krull was born in Rostock, studied in Berlin with Hertha Brämer, and made her stage debut in 1898 at the Plauen Stadttheater as Agathe in ''Der Freischütz''. From 1900 to 1912, she sang at the Dresden State Opera, where in 1901 she created the title role in Paderewski's ''Manru'' and Diemut in Richard Strauss' early opera '' Feuersnot''. Strauss, who had admired her dramatic qualities, then chose her to be the first Elektra. A year after its premiere in Dresden on 25 January 1909, she repeated the role at London's Royal Opera House. It was the first time a Strauss opera was performed in Britain. Krull sang regularly in several other German opera houses (Mannheim, Weimar, Leipzig, Cologne, Karlsruhe and Schwerin) as well as appearing in Brno (1905) and Prague (1907). Amongs ... [...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 * Krull–Azumaya theorem * Krull–Schmidt category * Krull–S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
