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Lasker-Noether Theorem
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by . The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodule ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring R and a two-sided ideal I in , a new ring, the quotient ring , is constructed, whose elements are the cosets of I in R subject to special + and \cdot operations. (Quotient ring notation almost always uses a fraction slash ""; stacking the ring over the ideal using a horizontal line as a separator is uncommon and generally avoided.) Quotient rings are distinct from the so-called "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization. Formal quotient ring construction Given a ring R and a two-sided ideal I in , we may define an e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients). In some older texts, the resultant is also called the eliminant. The resultant is widely used in number theory, either directly or through the discriminant, which is essentially the resultant of a polynomial and its derivative. The resultant of two polynomials with rational or polynomial coefficients may be computed efficiently on a computer. It is a basic tool of computer algebra, and is a built-in function of most computer algebra systems. It is used, among others, for cylindrical algebraic decomposition, integration of rational functions and drawing of curves defined by a bivariate polynomial equation. The resultant of ''n'' homogeneous polynomials in ''n'' variables (also called multivariate resultant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P''''n'', there should exist ''n'' − ''m'' homogeneous polynomials: :F_i(X_0,\cdots,X_n), 1\leq i\leq n - m, in the homogeneous coordinates ''X''''j'', which generate all other homogeneous polynomials that vanish on ''V''. Geometrically, each ''F''''i'' defines a hypersurface; the intersection of these hypersurfaces should be ''V''. The intersection of hypersurfaces will always have dimension at least ''m'', assuming that the field of scalars is an algebraically closed field such as the complex numbers. The question is essentially, can we get the dimension down to ''m'', with no extra points in the intersection? This condition is fairly hard to check as soon as the codimension . When then ''V'' is automatically a hypersurface and there is nothin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Height (ring Theory)
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal ''I'' in a polynomial ring ''R'' is the Krull dimension of ''R''/''I''. A field ''k'' has Krull dimension 0; more generally, ''k'' 'x''1, ..., ''x''''n''has Krull dimension ''n''. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent. There are several other ways that have been used to define the dimension of a ring. Most of them coi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Greatest Common Divisor
In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers. In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by the Euclidean algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant. The similarity between the integer GCD and the polynomial GCD allows extending to univariate polynomials all the properties that may be deduced from the Euclidean algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this provides information on the roots without c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5. The polynomial x^3 + 3 x^2 y + z^7 is not homogeneous, because the sum of exponents does not match from term to term. The function defined by a homogeneous polynomial is always a homogeneous function. An algebraic form, or simply form, is a function defined by a homogeneous polynomial.However, as some authors do not make a clear distinction between a polynomial and its associated function, the terms ''homogeneous polynomial'' and ''form'' are sometimes considered as synonymous. A binary form is a form in two variables. A ''form'' is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. A polynomial of degree 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers. 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 field of rational functions, fields of rational functions, algebraic function fields, algebraic number fields, and p-adic number, ''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 element (set), elements. The theory of fields proves that angle trisection and squaring the circle cannot be done with a compass and straighte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset P generated by a single element x \in P, which is to say the set of all elements less than or equal to x in P. The remainder of this article addresses the ring-theoretic concept. Definitions * A ''left principal ideal'' of R is a subset of R given by Ra = \ for some element a. * A ''right principal ideal'' of R is a subset of R given by aR = \ for some element a. * A ''two-sided principal ideal'' of R is a subset of R given by RaR = \ for some element a, namely, the set of all finite sums of elements of the form ras. While the definition for two-sided principal ideal may seem more complicated than for the one-sided principal ideals, it is necessary to ensure that the ideal remains closed under ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit (ring Theory)
In algebra, a unit or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng. Examples The multiplicative identity and its additive inverse are always units. More ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unique Factorization Domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non- unit element can be written as a product of irreducible elements, uniquely up to order and units. Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field. Unique factorization domains appear in the following chain of class inclusions: Definition Formally, a unique factorization domain is defined to be an integral domain ''R'' in which every non-zero element ''x'' of ''R'' which is not a unit can be written as a finite product of irreducible elements ''p''''i'' of ''R'': : ''x'' = ''p''1 ''p''2 � ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Prime Ideal
In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal prime ideals. Definition A prime ideal ''P'' is said to be a minimal prime ideal over an ideal ''I'' if it is minimal among all prime ideals containing ''I''. (Note: if ''I'' is a prime ideal, then ''I'' is the only minimal prime over it.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal. A minimal prime ideal over an ideal ''I'' in a Noetherian ring ''R'' is precisely a minimal associated prime (also called isolated prime) of R/I; this follows for instance from the primary decomposition of ''I''. Examples * In a commutative Artinian ring, every maximal ideal is a minimal prime ideal. * In an integral domain, the only minimal prime ideal is the zero ideal. * In the ring Z of integers, the minimal pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
