algebra over a field wikipedia - EAS
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Xem tất cả trên Wikipediahttps://en.wikipedia.org/wiki/Algebra_over_a_fieldIn mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and
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Definition
Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by · (that is, if x and y are any...
Xem thêmAlgebra homomorphisms
Given K-algebras A and B, a K-algebra homomorphism is a K-linear map f: A → B such that f(xy) = f(x) f(y) for all x, y in A. The space of all K-algebra homomorphisms between A and B is frequently written as...
Xem thêmAlgebras over fields come in many different types. These types are specified by insisting on some further axioms, such as
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Xem thêmFor algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A. Conversely, once a basis
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Xem thêmIn some areas of mathematics, such as commutative algebra, it is common to consider the more general concept of an algebra over a ring, where a commutative unital ring R replaces the field K. The only part of the definition that changes is that A is assumed to be an
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Xem thêmVăn bản Wikipedia theo giấy phép CC-BY-SAMục này có hữu ích không?Cảm ơn! Cung cấp thêm phản hồi- https://en.wikipedia.org/wiki/Talk:Algebra_over_a_field
- It's nice to see this separated out into its own thing as it should be.The problem is that there's really no good name for it."algebra over a field" isn't quite right, since algebras can be defined over any commutative ring with no change in the concept, and one does study these things with the same terminology and notation in (for example) algebraic geometry.But "algebra over a commu…
- https://en.wikipedia.org/wiki/Field_(mathematics)
e. In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
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- https://en.wikipedia.org/wiki/Algebra
Algebra (from Arabic الجبر (al-jabr) 'reunion of broken parts, bonesetting') is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.
Elementary algebra deals with the manipulation of variablesas if they were num…Wikipedia · Nội dung trong CC-BY-SA giấy phép - https://en.wikipedia.org/wiki/Algebraic_structure
Algebra over a ring (also R-algebra): a module over a commutative ring R, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and linearity with respect to multiplication by elements of R. The theory of an algebra over a field is especially well developed.
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algebra over a field - Wiktionary
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