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96 relations: Abelian group, Abstract algebra, Alfred Goldie, Alternative algebra, Annihilator (ring theory), Artin–Wedderburn theorem, Artin–Zorn theorem, Artinian ring, Ascending chain condition, Azumaya algebra, Øystein Ore, Cengage, Central simple algebra, Classification theorem, Clifford algebra, Commutative property, Commutative ring, Complex number, Derived algebraic geometry, Differential equation, Direct sum of modules, Division (mathematics), Division algebra, Division ring, Domain (ring theory), Emil Artin, Emmy Noether, Field (mathematics), Field of fractions, Finite field, Finite ring, Finite set, Finitely generated module, Galois cohomology, Group representation, Group ring, Harmonic analysis, Ideal (ring theory), Integral domain, Israel Nathan Herstein, Jacobson density theorem, Jacobson radical, John Wiley & Sons, Joseph Wedderburn, Kiiti Morita, Linear map, Localization of a ring, Mathematical Association of America, Mathematics, Matrix ring, ..., Maximal ideal, Microlocal analysis, Module (mathematics), Morita equivalence, Multiplicative inverse, Multiplicatively closed set, Nathan Jacobson, Noetherian ring, Noncommutative algebraic geometry, Noncommutative harmonic analysis, Noncommutative ring, Opposite ring, Ore condition, Paul Cohn, Preadditive category, Prime ring, Primitive ring, Principal ideal ring, Product ring, Quaternion, Real number, Richard Brauer, Ring (mathematics), Ring homomorphism, Ring theory, Scheme (mathematics), Semiprime ring, Semisimple algebra, Semisimple module, Simple algebra, Simple module, Simple ring, Springer Science+Business Media, Subdirect product, Superalgebra, Tensor product, Transactions of the American Mathematical Society, Uniform module, Unit (ring theory), Universal property, Vector space, Wedderburn's little theorem, Weyl algebra, William Rowan Hamilton, Zero element, Zero ring. Expand index (46 more) »
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.
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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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Alfred Goldie
Alfred William Goldie (10 December 1920, Coseley, Staffordshire – 8 October 2005, Barrow-in-Furness, Cumbria) was an English Mathematician.
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Alternative algebra
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.
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Annihilator (ring theory)
In mathematics, specifically module theory, the annihilator of a set is a concept generalizing torsion and orthogonality.
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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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Artin–Zorn theorem
In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field.
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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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Ascending chain condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.
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Azumaya algebra
In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field.
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Øystein Ore
Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics.
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Cengage
Cengage is an educational content, technology, and services company for the higher education, K-12, professional, and library markets worldwide.
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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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Classification theorem
In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?".
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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 property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.
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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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Derived algebraic geometry
Derived algebraic geometry (also called spectral algebraic geometry) is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by ring spectra in algebraic topology, whose higher homotopy accounts for the non-discreteness (e.g., Tor) of the structure sheaf.
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Differential equation
A differential equation is a mathematical equation that relates some function with its derivatives.
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Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module.
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Division (mathematics)
Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication.
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Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
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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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Domain (ring theory)
In mathematics, and more specifically in algebra, a domain is a nonzero ring in which implies or.
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Emil Artin
Emil Artin (March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
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Emmy Noether
Amalie Emmy NoetherEmmy is the Rufname, the second of two official given names, intended for daily use.
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Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
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Field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.
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Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
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Finite ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.
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Finite set
In mathematics, a finite set is a set that has a finite number of elements.
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Finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set.
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Galois cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups.
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Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.
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Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group.
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Harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).
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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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Integral domain
In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
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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.
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Jacobson density theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring.
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Jacobson radical
In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules.
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John Wiley & Sons
John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.
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Joseph Wedderburn
Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882, Forfar, Angus, Scotland – 9 October 1948, Princeton, New Jersey) was a Scottish mathematician, who taught at Princeton University for most of his career.
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Kiiti Morita
was a Japanese mathematician working in algebra and topology.
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Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
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Localization of a ring
In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring.
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Mathematical Association of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Matrix ring
In abstract algebra, a matrix ring is any collection of matrices over some ring R that form a ring under matrix addition and matrix multiplication.
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Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.
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Microlocal analysis
In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations.
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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
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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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Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.
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Multiplicatively closed set
In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset S of a ring R such that the following two conditions hold.
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Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.
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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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Noncommutative algebraic geometry
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative stack quotients).
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Noncommutative harmonic analysis
In mathematics, noncommutative harmonic analysis is the field in which results from Fourier analysis are extended to topological groups that are not commutative.
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Noncommutative ring
In mathematics, more specifically abstract algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exists a and b in R with a·b ≠ b·a.
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Opposite ring
In algebra, the opposite of a ring is another ring with the same elements and addition operation, but with the multiplication performed in the reverse order.
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Ore condition
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring.
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Paul Cohn
Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986-9, and author of many textbooks on algebra.
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Preadditive category
In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups.
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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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Principal ideal ring
In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring.
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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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Richard Brauer
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician.
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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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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.
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Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.
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Semiprime ring
In ring theory, a branch of mathematics, semiprime ideals and semiprime rings are generalizations of prime ideals and prime rings.
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Semisimple algebra
In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical).
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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 algebra
In mathematics, specifically in ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the multiplication operation is not zero (that is, there is some a and some b such that). The second condition in the definition precludes the following situation; consider the algebra with the usual matrix operations: \left\ This is a one-dimensional algebra in which the product of any two elements is zero.
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Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that have no non-zero proper submodules.
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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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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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Subdirect product
In mathematics, especially in the areas of abstract algebra known as universal algebra, group theory, ring theory, and module theory, a subdirect product is a subalgebra of a direct product that depends fully on all its factors without however necessarily being the whole direct product.
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Superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra.
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Tensor product
In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.
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Transactions of the American Mathematical Society
The Transactions of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society.
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Uniform module
In abstract algebra, a module is called a uniform module if the intersection of any two nonzero submodules is nonzero.
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Unit (ring theory)
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.
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Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Wedderburn's little theorem
In mathematics, Wedderburn's little theorem states that every finite domain is a field.
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Weyl algebra
In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), namely expressions of the form More precisely, let F be the underlying field, and let F be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F. ∂X is the derivative with respect to X. The algebra is generated by X and ∂X.
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William Rowan Hamilton
Sir William Rowan Hamilton MRIA (4 August 1805 – 2 September 1865) was an Irish mathematician who made important contributions to classical mechanics, optics, and algebra.
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Zero element
In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures.
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Zero ring
In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element.
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References
[1] https://en.wikipedia.org/wiki/Noncommutative_ring
