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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] |
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] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of module (mathematics), modules and Syzygy (mathematics), syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariant (mathematics), invariants of ring (mathematics), rings, modules, topological spaces, and other "tangible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finitely Generated Module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group. Definition The left ''R''-module ''M'' is finitely generated if there exist ''a''1, ''a''2, ..., ''a''''n'' in ''M'' such that for any ''x'' in ''M'', there exist ''r''1, ''r''2, ..., ''r''''n'' in ''R'' with ''x'' = ''r''1''a''1 + ''r''2''a''2 + ... + ''r''''n''''a''''n''. The set is referred to as a gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...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] |
Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are function (mathematics), functions on the group of chain (algebraic topology), chains in homology theory. From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and abstract algebra, algebra. The terminology tends to hide the fact that cohomology, a Covariance and contravariance of functors, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ext Functor
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The group cohomology, cohomology of groups, Lie algebra cohomology, Lie algebras, and Hochschild cohomology, associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies group extension, extensions of one module (mathematics), module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring (mathematics), ring, Ext was defined by Henri Cartan and Eilenberg in their 1956 book ''Homological Algebra''. Definition Let R be a ring and let R\text be the category (mathematics ... [...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] |
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] |
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] |
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] |
Depth (ring Theory)
In commutative algebra, commutative and homological algebra, homological algebra, depth is an important invariant of ring (mathematics), rings and module (mathematics), modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian ring, Noetherian local ring. In this case, the depth of a module is related with its projective dimension by the Auslander–Buchsbaum formula. A more elementary property of depth is the inequality : \mathrm(M) \leq \dim(M), where \dim M denotes the Krull dimension of the module M. Depth is used to define classes of rings and modules with good properties, for example, Cohen-Macaulay rings and modules, for which equality holds. Definition Let R be a commutative ring, I an ideal of R and M a finitely generated module, finitely generated R-module with the property that I M is properly contained in M. (That is, some elements of M are not in I M.) Then the I-depth of M, also com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
