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Perfect Ideal
In commutative algebra, a perfect ideal is a proper ideal I in a Noetherian ring R such that its grade equals the projective dimension of the associated quotient ring. \textrm(I)=\textrm\dim(R/I). A perfect ideal is unmixed. For a regular local ring R a prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ... I is perfect if and only if R/I is Cohen-Macaulay. The notion of perfect ideal was introduced in 1913 by Francis Sowerby Macaulay in connection to what nowadays is called a Cohen-Macaulay ring, but for which Macaulay did not have a name for yet. As Eisenbud and Gray point out, Macaulay's original definition of perfect ideal I coincides with the modern definition when I is a homogeneous ideal in a polynomial ring, but may differ otherwise. Macaulay used Hilbert ... [...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 ideal (ring theory), ideals, and module (mathematics), 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 number, ''p''-adic integers. Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts. 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. Several concepts of commutative algebras have been developed in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. Formally, every increasing sequence I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an n such that I_=I_=\cdots. Equivalently, a ring is left-Noetherian (respectively right-Noetherian) if every left ideal (respectively right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on the Noetherian property ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grade (ring Theory)
In commutative and homological algebra, the grade of a finitely generated module M over a Noetherian ring R is a cohomological invariant defined by vanishing of Ext-modules \textrm\,M=\textrm_R\,M=\inf\left\. For an ideal I\triangleleft R the grade is defined via the quotient ring viewed as a module over R \textrm\,I=\textrm_R\,I=\textrm_R\,R/I=\inf\left\. The grade is used to define perfect ideals. In general we have the inequality \textrm_R\,I\leq\textrm\dim(R/I) where the projective dimension is another cohomological invariant. The grade is tightly related to the depth, since \textrm_R\,I=\textrm_(R). Under the same conditions on R, I and M as above, one also defines the M-grade of I as \textrm_M\,I=\inf\left\. This notion is tied to the existence of maximal M-sequences contained in I of length \textrm_M\,I. References {{abstract-algebra-stub Ring theory Homological algebra Commutative algebra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Dimension
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various equivalent characterizations of these modules appear below. Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains. However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem). Projective modules were first introduced in 1956 in the influential book ''Homological Algebra'' by Henri Cartan and Samuel Eilenberg. Definitions Lifting property The usual category theoretical definition is in terms of the property of ''lifting'' that carries over from free to projective modules: a module ''P'' is projective if and only if for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unmixed Ideal
''Unmixed'' is the second album by house production duo Freemasons. It was released on 29 October 2007, with the lead single "Uninvited" being released one week earlier. The album is unique, as unlike most DJ/Producers, the tracks are all unmixed full length versions. In addition, producer samples packs created by the Freemasons were included on a data portion of the album. Track listing Samples, covers and interpolations * "Uninvited" is a cover of Alanis Morissette's song. * "If" is a cover of Jackie Moore's song, released in 1973 on her album ''Sweet Charlie Babe''. * "Nothing But A Heartache" is a cover of The Flirtations' song. * "Love on My Mind" incorporates lyrics from Tina Turner's song " When the Heartache Is Over" and samples Jackie Moore's song "This Time Baby". * "Watchin'" incorporates elements from Deborah Cox's song "It's Over Now", released in 1998 on her album '' One Wish''. * "Love Don't Live Here Anymore" is a cover of Rose Royce's song. * "Pacifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Local Ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maximal ideal \mathfrak, and suppose a_1,\cdots,a_n is a minimal set of generators of \mathfrak. Then Krull's principal ideal theorem implies that n\geq\dim A, and A is regular whenever n=\dim A. The concept is motivated by its geometric meaning. A point x on an algebraic variety X is nonsingular (a smooth point) if and only if the local ring \mathcal_ of germs at x is regular. (See also: regular scheme.) Regular local rings are ''not'' related to von Neumann regular rings. For Noetherian local rings, there is the following chain of inclusions: Characterizations There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if A is a Noetherian local ring with maximal ideal \mathfrak, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Ideal
In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all the multiple (mathematics), multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary ideal, primary and semiprime ideal, semiprime. Prime ideals for commutative rings Definition An ideal (ring theory), ideal of a commutative ring is prime if it has the following two properties: * If and are two elements of such that their product is an element of , then is in or is in , * is not the whole ring . This generalizes the following property of prime numbers, known as Euclid's lemma: if is a prime number and if divides a product of two integers, then divides or divides . We can therefore say :A positive integer is a prime number if and only if n\Z is a prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Francis Sowerby Macaulay
Francis Sowerby Macaulay FRS (11 February 1862, Witney – 9 February 1937, Cambridge) was an English mathematician who made significant contributions to algebraic geometry. He is known for his 1916 book ''The Algebraic Theory of Modular Systems'' (an old term for ideals), which greatly influenced the later course of commutative algebra. Cohen–Macaulay rings, Macaulay duality, the Macaulay resultant and the Macaulay and Macaulay2 computer algebra systems are named for Macaulay. Macaulay was educated at Kingswood School and graduated with distinction from St John's College, Cambridge. He taught the top mathematics class in St Paul's School in London from 1885 to 1911. His students included J. E. Littlewood and G. N. Watson. In 1928 Macaulay was elected Fellow of the Royal Society Fellowship of the Royal Society (FRS, ForMemRS and HonFRS) is an award granted by the Fellows of the Royal Society of London to individuals who have made a "substantial contribution t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Function
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which measure the growth of the dimension of the homogeneous components of the algebra. These notions have been extended to filtered algebras, and graded or filtered modules over these algebras, as well as to coherent sheaves over projective schemes. The typical situations where these notions are used are the following: * The quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. * The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. * The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial. The Hilbert series of an algebra or a module is a special case of the Hilbert–Poincaré series of a graded vector space. The Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
