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In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.
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mathematics and associative
In mathematics, the associative property is a property of some binary operations.
Associative operations are abundant in mathematics ; in fact, many algebraic structures ( such as semigroups and categories ) explicitly require their binary operations to be associative.
In some areas of mathematics, associative algebras are typically assumed to have a multiplicative unit, denoted 1.
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space.
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element.
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.
In mathematics, Clifford algebras are a type of associative algebra.
* Incidence algebra, associative algebra used in combinatorics, a branch of mathematics
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for any locally finite partially ordered set
In mathematics, a *- ring is an associative ring with a map *: A → A which is an antiautomorphism and an involution.
In mathematics, coalgebras or cogebras are structures that are dual ( in the sense of reversing arrows ) to unital associative algebras.
In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra, such that these structures are compatible.
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an ( unital associative ) algebra and a ( counital coassociative ) coalgebra, with these structures ' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz ' law ; that is, the bracket is also a derivation.
In ring theory and related areas of mathematics a central simple algebra ( CSA ) over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K. In other words, any simple algebra is a central simple algebra over its center.
In mathematics, a bicategory is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not ( strictly ) associative, but only associative up to an isomorphism.
In mathematics, the symmetric algebra S ( V ) ( also denoted Sym ( V )) on a vector space V over a field K is the free commutative unital associative algebra over K containing V.
In mathematics, inversive ring geometry is the extension of the concepts of projective line, homogeneous coordinates, projective transformations, and cross-ratio to the context of associative rings, concepts usually built upon rings that happen to be fields.
In noncommutative geometry and related branches of mathematics, cyclic homology and cyclic cohomology are certain ( co ) homology theories for associative algebras which generalize the de Rham ( co ) homology of manifolds.
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories.
In mathematics, the Moyal product, named after José Enrique Moyal, is perhaps the best-known example of a phase-space star product: an associative, non-commutative product, ★, on the functions on ℝ < sup > 2n </ sup >, equipped with its Poisson bracket ( with a generalization to symplectic manifolds below ).
mathematics and algebra
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.
In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.
Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.
The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics ( for example Venn diagrams and symbolic reasoning about their Boolean algebra ), and the everyday usage of set theory concepts in most contemporary mathematics.
Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.
General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later ; it is now applied throughout mathematics.
It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis.
Mathematics used in rendering includes: linear algebra, calculus, numerical mathematics, signal processing, and Monte Carlo methods.
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.
The invention of Cartesian coordinates in the 17th century by René Descartes ( Latinized name: Cartesius ) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.
Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics.
Diophantus ' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra.
In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.
Elementary algebra introduces the basic rules and operations of algebra, one of the main branches of mathematics.
Elementary algebra is typically taught to secondary school students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic as " algebra ".
mathematics and is
This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.
Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.
But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.
Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.
In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.
So, too, is the mathematical competence of a college graduate who has majored in mathematics.
The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.
In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.
The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.
In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.
The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.
In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.
The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.
:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.
Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.
There is no prize awarded for mathematics, but see Abel Prize.
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