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Algebra Homomorphism
A homomorphism between two algebras, A and B, over a field (or ring) K, is a map F : A → B such that for all k in K and x, y in A,[1][2]F(kx) = kF(x) F(x + y) = F(x) + F(y) F(xy) = F(x)F(y)If F is bijective then F is said to be an isomorphism between A and B. A common abbreviation for "homomorphism between algebras" is "algebra homomorphism" or "algebra map"
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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. For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras
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Graduate Texts In Mathematics
Graduate Texts in Mathematics
Mathematics
(GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag
Springer-Verlag
mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics
Mathematics
series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level.Contents1 List of books 2 See also 3 Notes 4 External linksList of books[edit]Introduction to Axiomatic Set Theory, Gaisi Takeuti, Wilson M
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
(album), a 1992 album by Vesta Williams "Special" (Garbage song), 1998 "Special
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International Standard Book Number
"ISBN" redirects here. For other uses, see ISBN (other).International Standard Book
Book
NumberA 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar codeAcronym ISBNIntroduced 1970; 48 years ago (1970)Managing organisation International ISBN AgencyNo. of digits 13 (formerly 10)Check digit Weighted sumExample 978-3-16-148410-0Website www.isbn-international.orgThe International Standard Book
Book
Number (ISBN) is a unique[a][b] numeric commercial book identifier. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.[1] An ISBN is assigned to each edition and variation (except reprintings) of a book. For example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, and 10 digits long if assigned before 2007
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John Wiley & Sons
John Wiley & Sons, Inc., also referred to as Wiley (NYSE: JW.A), is a global publishing company that specializes in academic publishing. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services,[3] training materials, and educational materials for undergraduate, graduate, and continuing education students.[4] Founded in 1807, Wiley is also known for publishing For Dummies. As of 2015, the company had 4,900 employees and a revenue of $1.8 billion.[1]Contents1 History1.1 High-growth and emerging markets 1.2 Strategic acquisition and divestiture2 Governance and operations 3 Brands and partnerships 4 Worldwide partnership with Christian H. Cooper 5 Current initiatives5.1 Higher education 5.2 Medicine 5.3 Architecture and design6 Wiley Online Library 7 Corporate culture 8 Apple controversy 9 Kirtsaeng v
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Spectrum Of A Ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec ⁡ ( R ) displaystyle operatorname Spec (R) , is the set of all prime ideals of R. It is commonly augmented with the Zariski topology
Zariski topology
and with a structure sheaf, turning it into a locally ringed space. A locally ringed space of this form is called an affine scheme.Contents1 Zariski topology 2 Sheaves and schemes 3 Functorial perspective 4 Motivation from algebraic geometry 5 Examples 6 Not-Affine Examples 7 Global or relative Spec 8 Example of relative Spec 9 Representation theory
Representation theory
perspective 10 Functional analysis perspective 11 Generalizations 12 See also 13 References 14 External linksZariski topology[edit] For any ideal I of R, define V I displaystyle V_ I to be the set of prime ideals containing I
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Skolem–Noether Theorem
In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme (German: On the theory of associative number systems) and later rediscovered by Emmy Noether.Contents1 Statement 2 Proof 3 Notes 4 ReferencesStatement[edit] In a general formulation, let A and B be simple unitary rings, and let k be the centre of B. Notice that k is a field since given x nonzero in k, the simplicity of B implies that the nonzero two-sided ideal BxB = (x) is the whole of B, and hence that x is a unit. If the dimension of B over k is finite, i.e
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Springer Science+Business Media
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.[1] Springer also hosts a number of scientific databases, including SpringerLink, Springer Protocols, and SpringerImages
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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 ab ≠ 0). The second condition in the definition precludes the following situation; consider the algebra with the usual matrix operations: [ 0 α 0 0 ] α ∈ C displaystyle left left. begin bmatrix 0&alpha \0&0\end bmatrix ,right,alpha in mathbb C right This is a one-dimensional algebra in which the product of any two elements is zero
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Group Of Units
In mathematics, an invertible element or a unit in a (unital) ring R is any element u that has an inverse element in the multiplicative monoid of R, i.e. an element v such thatuv = vu = 1R, where 1R is the multiplicative identity.[1][2]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. It never contains the element 0 (except in the case of the zero ring), and is therefore not closed under addition; its complement however might be a group under addition, which happens if and only if the ring is a local ring. The term unit is also used to refer to the identity element 1R of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. For this reason, some authors call 1R "unity" or "identity", and say that R is a "ring with unity" or a "ring with identity" rather than a "ring with a unit". The multiplicative identity 1R and its opposite −1R are always units
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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. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures. Subalgebra can be a subset of both cases.Contents1 Subalgebras for algebras over a ring or field1.1 Example2 Subalgebras in universal algebra2.1 Example3 ReferencesSubalgebras for algebras over a ring or field[edit] A subalgebra of an algebra over a commutative ring or field is a vector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. to associative algebras or to Lie algebras
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Lagrange Interpolation
In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of points ( x j , y j ) displaystyle (x_ j ,y_ j ) with no two x j displaystyle x_ j values equal, the Lagrange polynomial
Lagrange polynomial
is the polynomial of lowest degree that assumes at each value x j displaystyle x_ j the corresponding value y j displaystyle y_ j (i.e. the functions coincide at each point)
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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. More explicitly, if R and S are rings, then a ring homomorphism is a function f : R → S such that[1][2][3][4][5][6]f(a + b) = f(a) + f(b) for all a and b in R f(ab) = f(a) f(b) for all a and b in R f(1R) = 1S.(Additive inverses and the additive identity are part of the structure too, but it is not necessary to require explicitly that they too are respected, because these conditions are consequences of the three conditions above. On the other hand, neglecting to include the condition f(1R) = 1S would cause several of the properties below to fail.) If R and S are rngs (also known as pseudo-rings, or non-unital rings), then the natural notion[7] is that of a rng homomorphism, defined as above except without the third condition f(1R) = 1S
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Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, wherein the corresponding scalars are the elements of an arbitrary given ring (with identity) and a multiplication (on the left and/or on the right) is defined between elements of the ring and elements of the module. Thus, a module, like a vector space, is an additive abelian group; a product is defined between elements of the ring and elements of the module that is distributive over the addition operation of each parameter and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups
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Bijective
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements
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