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Rees Decomposition
In commutative algebra, a Rees decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by . Definition Suppose that a ring ''R'' is a quotient of a polynomial ring ''k'' 'x''1,...over a field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ... by some homogeneous ideal. A Rees decomposition of ''R'' is a representation of ''R'' as a direct sum (of vector spaces) : R = \bigoplus_\alpha \eta_\alpha k theta_1,\ldots,\theta_ where each ''η''''α'' is a homogeneous element and the ''d'' elements ''θ''''i'' are a homogeneous system of parameters for ''R'' and ''η''''α''''k'' 'θ''''f''''α''+1,...,''θ''''d''⊆ ''k'' 'θ''1, ''θ''''f''''α'' See also * Stanley decomposition * Hironaka decomposition References * * Commutative algebra [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers \mathbb; and ''p''-adic integers. Commutative algebra is the main technical tool in the local study of schemes. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras. Overview Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry. In algebraic number theory, the rings of algebraic integers are Dedekind rings, which constitute therefore an important class of commutative rings. Considerations related to modular arithmetic have led to the no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
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. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stanley Decomposition
In commutative algebra, a Stanley decomposition is a way of writing a ring in terms of polynomial subrings. They were introduced by . Definition Suppose that a ring ''R'' is a quotient of a polynomial ring ''k'' 'x''1,...over a field by some ideal. A Stanley decomposition of ''R'' is a representation of ''R'' as a direct sum (of vector spaces) : R = \bigoplus_\alpha x_\alpha k(X_\alpha) where each ''x''''α'' is a monomial and each ''X''''α'' is a finite subset of the generators. See also *Rees decomposition *Hironaka decomposition In mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this in ... References * * Commutative algebra {{commutative-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hironaka Decomposition
In mathematics, a Hironaka decomposition is a representation of an algebra over a field as a finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this in his unpublished master's thesis at Kyoto University . Hironaka's criterion , sometimes called miracle flatness, states that a local ring ''R'' that is a finitely generated module over a regular Noetherian local ring ''S'' is Cohen–Macaulay if and only if it is a free module over ''S''. There is a similar result for rings that are graded over a field rather than local. Explicit decomposition of an invariant algebra Let V be a finite-dimensional vector space over an algebraically closed field of characteristic zero, K , carrying a representation of a group G, and consider the polynomial algebra on V, K /math>. The algebra K carries a grading with (K _0 = K , which is inherited by the invariant subalgebra : K G = \. A famous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
