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63 relations: *-algebra, Absolute convergence, Abstract algebra, Algebra homomorphism, Algebra over a field, Angular momentum, Artin–Wedderburn theorem, Artinian ring, Associative algebra, Banach algebra, Bicomplex number, Biquaternion, C*-algebra, Center (ring theory), Central simple algebra, Clifford algebra, Commutative ring, Complex number, Diagonal matrix, Dimension (vector space), Division ring, Endomorphism ring, Free module, Frobenius algebra, Gelfand–Naimark theorem, General linear group, Generic matrix ring, Group (mathematics), Hurwitz's theorem (composition algebras), Ideal (ring theory), Idempotent (ring theory), Identity matrix, Invertible matrix, Matrix (mathematics), Matrix addition, Matrix multiplication, Matrix unit, Morita equivalence, Nilpotent, Noetherian ring, Partial isometry, Pauli matrices, Prime ring, Primitive ring, Product ring, Quaternion, Real number, Ring (mathematics), Ring homomorphism, Semiring, ..., Semisimple module, Simple ring, Split-biquaternion, Split-complex number, Split-quaternion, Springer Science+Business Media, Stably finite ring, Subalgebra, Transpose, Triangular matrix, William Kingdon Clifford, Zero divisor, 2 × 2 real matrices. Expand index (13 more) »
*-algebra
In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings and, where is commutative and has the structure of an associative algebra over.
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Absolute convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.
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Abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.
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Algebra homomorphism
A homomorphism between two associative algebras, A and B, over a field (or commutative ring) K, is a function F\colon A\to B such that for all k in K and x, y in A,.
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Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
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Angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum.
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Artin–Wedderburn theorem
In algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras.
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Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals.
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Associative algebra
In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.
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Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm.
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Bicomplex number
In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate (w,z)^*.
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Biquaternion
In abstract algebra, the biquaternions are the numbers, where, and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group.
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C*-algebra
C∗-algebras (pronounced "C-star") are an area of research in functional analysis, a branch of mathematics.
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Center (ring theory)
In algebra, the center of a ring R is the subring consisting of the elements x such that xy.
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Central simple algebra
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.
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Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.
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Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.
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Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.
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Endomorphism ring
In abstract algebra, the endomorphism ring of an abelian group X, denoted by End(X), is the set of all endomorphisms of X (i.e., the set of all homomorphisms of X into itself) endowed with an addition operation defined by pointwise addition of functions and a multiplication operation defined by function composition.
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Free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements.
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Frobenius algebra
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.
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Gelfand–Naimark theorem
In mathematics, the Gelfand–Naimark theorem states that an arbitrary C*-algebra A is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space.
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General linear group
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
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Generic matrix ring
In algebra, a generic matrix ring is a sort of a universal matrix ring.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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Hurwitz's theorem (composition algebras)
In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic form.
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Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
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Idempotent (ring theory)
In abstract algebra, more specifically in ring theory, an idempotent element, or simply an idempotent, of a ring is an element a such that.
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Identity matrix
In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.
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Invertible matrix
In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Matrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together.
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Matrix multiplication
In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.
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Matrix unit
In mathematics, a matrix unit is an idealisation of the concept of a matrix, with a focus on the algebraic properties of matrix multiplication.
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Morita equivalence
In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties.
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Nilpotent
In mathematics, an element, x, of a ring, R, is called nilpotent if there exists some positive integer, n, such that xn.
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Noetherian ring
In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any chain of left (or right) ideals: there exists an n such that: Noetherian rings are named after Emmy Noether.
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Partial isometry
In functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel.
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Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian and unitary.
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Prime ring
In abstract algebra, a nonzero ring R is a prime ring if for any two elements a and b of R, arb.
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Primitive ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module.
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Product ring
In mathematics, it is possible to combine several rings into one large product ring.
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Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
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Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.
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Semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.
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Semisimple module
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts.
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Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself.
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Split-biquaternion
In mathematics, a split-biquaternion is a hypercomplex number of the form where w, x, y, and z are split-complex numbers and i, j, and k multiply as in the quaternion group.
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Split-complex number
In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written z.
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Split-quaternion
In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Stably finite ring
In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A, B of the same size over R, AB.
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Subalgebra
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
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Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At).
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Triangular matrix
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix.
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William Kingdon Clifford
William Kingdon Clifford FRS (4 May 1845 – 3 March 1879) was an English mathematician and philosopher.
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Zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.
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2 × 2 real matrices
In mathematics, the associative algebra of real matrices is denoted by M(2, R).
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References
[1] https://en.wikipedia.org/wiki/Matrix_ring
