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Binomial Ring
In mathematics, a binomial ring is a commutative ring whose additive group is torsion-free and contains all binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...s :\binom = \frac for ''x'' in the ring and ''n'' a positive integer. Binomial rings were introduced by . showed that binomial rings are essentially the same as λ-rings for which all Adams operations are the identity. References * * *{{citation, mr=2649360 , last=Yau, first= Donald , title=Lambda-rings, publisher= World Scientific Publishing Co. Pte. Ltd., place= Hackensack, NJ, year= 2010 , isbn= 978-981-4299-09-1 , url=https://books.google.com/books?id=d7vKnjxyvxQC Ring theory ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three axioms hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of symmetries and geometric transformations: The symmetries of an object form a group, called the symmetry group of the ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion (algebra)
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element. This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements. This terminology applies to abelian groups (with "module" and "submodule" replaced by " group" and "subgroup"). This is allowed by the fact that the abelian groups are the modules over the ring of integers (in fact, this is the origin of the terminology, that has been introduced for abelian groups before being generalized to modules). In the case of groups that are noncommutative, a ''torsion element'' is an element of finite order. Contrary to the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lambda Ring
Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the Dental, alveolar and postalveolar lateral approximants, voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician alphabet, Phoenician Lamed . Lambda gave rise to the Latin script, Latin L and the Cyrillic script, Cyrillic El (Cyrillic), El (Л). The ancient Alexandrine grammarians, grammarians and dramatists give evidence to the pronunciation as () in Classical Greek times. In Modern Greek, the name of the letter, Λάμδα, is pronounced . In Epichoric alphabets, early Greek alphabets, the shape and orientation of lambda varied. Most variants consisted of two straight strokes, one longer than the other, connected at their ends. The angle might be in the upper-left, lower-left ("Western" alphabets) or top ("Eastern" alphabets). Other variants had a vertical line with a horizontal or sloped stroke running to the right. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adams Operation
In mathematics, an Adams operation, denoted ψ''k'' for natural numbers ''k'', is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. The basic idea is to implement some fundamental identities in symmetric function theory, at the level of vector bundles or other representing object in more abstract theories. Adams operations can be defined more generally in any λ-ring. Adams operations in K-theory Adams operations ψ''k'' on K theory (algebraic or topological) are characterized by the following properties. # ψ''k'' are ring homomorphisms. # ψ''k''(l)= lk if l is the class of a line bundle. # ψ''k'' are functorial. The fundamental idea is that for a vector bundle ''V'' on a topological space ''X'', there is an analogy between Adams operators and exterior powers, in which :ψ''k''(''V'') is to Λ''k''(''V'') as :the power sum Σ α''k'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
