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Jacobson's Conjecture
In abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring. It has only been proven for special types of Noetherian rings, so far. Examples exist to show that the conjecture can fail when the ring is not Noetherian on a side, so it is absolutely necessary for the ring to be two-sided Noetherian. The conjecture is named for the algebraist Nathan Jacobson who posed the first version of the conjecture. Statement For a ring ''R'' with Jacobson radical ''J'', the nonnegative powers J^n are defined by using the product of ideals. :''Jacobson's conjecture:'' In a right-and-left Noetherian ring, \bigcap_J^n=\. In other words: "The only element of a Noetherian ring in all powers of ''J'' is 0." The original conjecture posed by Jacobson in 1956. As cited by . asked about noncommutative one-sided Noetherian rings, however Israel Nathan Herstein produced a counterexample in 1965, and soon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The National Academy Of Sciences Of The United States Of America
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2021 impact factor of 12.779. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the mass media, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are ava ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Layer Condition
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of Units ( SI) is more precise:The second ..is defined by taking the fixed numerical value of the caesium frequency, Δ''ν''Cs, the unperturbed ground-state hyperfine transition frequency of the caesium 133 atom, to be when expressed in the unit Hz, which is equal to s−1. This current definition was adopted in 1967 when it became feasible to define the second based on fundamental properties of nature with caesium clocks. Because the speed of Earth's rotation varies and is slowing ever so slightly, a leap second is added at irregular intervals to civil time to keep clocks in sync with Earth's rotation. Uses Analog clocks and watches often have ... [...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] |
Fully Bounded Noetherian Ring
Fully () is a municipality in the district of Martigny in the canton of Valais in Switzerland. History Fully is first mentioned in the 11th Century as ''Fuliacum''. Geography Fully has an area, , of . Of this area, 30.5% is used for agricultural purposes, while 27.7% is forested. Of the rest of the land, 5.7% is settled (buildings or roads) and 36.2% is unproductive land. It is on the right bank of the river Rhône and is well known for its wines. Fully is just the administrative name of a group of several villages of Vers-l'Eglise (sometimes called Fully), Branson, Châtaignier and Randonnaz along with a number of hamlets. The natural reserve of Les Follatères (shared between Fully and Dorénaz), located on the south facing slopes above the Rhône elbow, has a variety of animal and plant species normally uncommon in Switzerland. Fully-suisse.jpg, Fully at the foot of the Grand Chavalard Colourful lizard - panoramio.jpg, Green lizard Cactuses.jpg, Cactuses (Opuntia) Coa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull Intersection Theorem
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] |
Principal Ideal Domain
In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot. Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If and are elements of a PID without common divisors, then every element of the PID can be written in the form . Principal ideal domains are noetherian, they are integra ... [...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] |
Israel Nathan Herstein
Israel Nathan Herstein (March 28, 1923 – February 9, 1988) was a mathematician, appointed as professor at the University of Chicago in 1951. He worked on a variety of areas of algebra, including ring theory, with over 100 research papers and over a dozen books. Education and career Herstein was born in Lublin, Poland, in 1923. His family emigrated to Canada in 1926, and he grew up in a harsh and underprivileged environment where, according to him, "you either became a gangster or a college professor." During his school years he played football, ice hockey, golf, tennis, and pool. He also worked as a steeplejack and as a barker at a fair. He received his B.S. degree from the University of Manitoba and his M.A. from the University of Toronto. He received his Ph.D from Indiana University in 1948. His advisor was Max Zorn. He held positions at the University of Kansas, Ohio State University, University of Pennsylvania, and Cornell University before permanently settling at the Uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glasgow Mathematical Journal
The ''Glasgow Mathematical Journal'' is a mathematics journal that publishes original research papers in any branch of pure and applied mathematics. It covers a wide variety of research areas, which in recent issues have included ring theory, group theory, functional analysis, combinatorics, differential equations, differential geometry, number theory, algebraic topology, and the application of such methods in applied mathematics. The editor-in-chief is currently I. A. B. Strachan (University of Glasgow , image = UofG Coat of Arms.png , image_size = 150px , caption = Coat of arms Flag , latin_name = Universitas Glasguensis , motto = la, Via, Veritas, Vita , ...). References {{Mathematics in the United Kingdom Mathematics education in the United Kingdom Mathematics journals Cambridge University Press academic journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
