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Weak Dimension
In abstract algebra, the weak dimension of a nonzero right module ''M'' over a ring ''R'' is the largest number ''n'' such that the Tor group \operatorname_n^R(M,N) is nonzero for some left ''R''-module ''N'' (or infinity if no largest such ''n'' exists), and the weak dimension of a left ''R''-module is defined similarly. The weak dimension was introduced by . The weak dimension is sometimes called the flat dimension as it is the shortest length of a resolution of the module by flat modules. The weak dimension of a module is at most equal to its projective dimension. The weak global dimension of a ring is the largest number ''n'' such that \operatorname_n^R(M,N) is nonzero for some right ''R''-module ''M'' and left ''R''-module ''N''. If there is no such largest number ''n'', the weak global dimension is defined to be infinite. It is at most equal to the left or right global dimension of the ring ''R''. Examples *The module \Q of rational numbers over the ring \Z of integers ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Dimension
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module ''Q'' that shares certain desirable properties with the Z-module Q of all rational numbers. Specifically, if ''Q'' is a submodule of some other module, then it is already a direct summand of that module; also, given a submodule of a module ''Y'', then any module homomorphism from this submodule to ''Q'' can be extended to a homomorphism from all of ''Y'' to ''Q''. This concept is dual to that of projective modules. Injective modules were introduced in and are discussed in some detail in the textbook . Injective modules have been heavily studied, and a variety of additional notions are defined in terms of them: Injective cogenerators are injective modules that faithfully represent the entire category of modules. Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in the derived category. Injective h ... [...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] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, ''The Daily Princetonian'', and later added book publishing to it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Matrix Ring
In algebra, a triangular matrix ring, also called a triangular ring, is a ring constructed from two rings and a bimodule. Definition If T and U are rings and M is a \left(U,T\right)-bimodule, then the triangular matrix ring R:=\left beginT&0\\M&U\\\end\right/math> consists of 2-by-2 matrices of the form \left begint&0\\m&u\\\end\right/math>, where t\in T,m\in M, and u\in U, with ordinary matrix addition and matrix multiplication as its operations. References *{{Citation , last1=Auslander , first1=Maurice , last2=Reiten , first2=Idun , last3=Smalø , first3=Sverre O. , title=Representation theory of Artin algebras , origyear=1995 , url=https://books.google.com/books?isbn=0521599237 , publisher=Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ... , ... [...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. That is, 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 such that: I_=I_=\cdots. Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. 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 Noetherian property (for example, the Laskerâ ... [...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] |
Product Of Rings
In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in the category of rings. Since direct products are defined up to an isomorphism, one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings. For example, the Chinese remainder theorem may be stated as: if and are coprime integers, the quotient ring \Z/mn\Z is the product of \Z/m\Z and \Z/n\Z. Examples An important example is Z/''n''Z, the ring of integers modulo ''n''. If ''n'' is written as a product of prime powers (see Fundamental theorem of arithmetic), :n=p_1^ p_2^\cdots\ p_k^, where the ''pi'' are distinct primes, then Z/''n''Z is naturally isomorphic to the product :\mathbf/p_1^\mathbf \ \times \ \mathbf/p_2^\mathbf \ \times \ \cdots \ \times \ \mathbf/p_k^\mathbf. This f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Von Neumann Regular Ring
In mathematics, a von Neumann regular ring is a ring ''R'' (associative, with 1, not necessarily commutative) such that for every element ''a'' in ''R'' there exists an ''x'' in ''R'' with . One may think of ''x'' as a "weak inverse" of the element ''a;'' in general ''x'' is not uniquely determined by ''a''. Von Neumann regular rings are also called absolutely flat rings, because these rings are characterized by the fact that every left ''R''-module is flat. Von Neumann regular rings were introduced by under the name of "regular rings", in the course of his study of von Neumann algebras and continuous geometry. Von Neumann regular rings should not be confused with the unrelated regular rings and regular local rings of commutative algebra. An element ''a'' of a ring is called a von Neumann regular element if there exists an ''x'' such that .Kaplansky (1972) p.110 An ideal \mathfrak is called a (von Neumann) regular ideal if for every element ''a'' in \mathfrak there exists an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prüfer Domain
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer. Examples The ring of entire functions on the open complex plane C form a Prüfer domain. The ring of integer valued polynomials with rational coefficients is a Prüfer domain, although the ring \mathbb /math> of integer polynomials is not . While every number ring is a Dedekind domain, their union, the ring of algebraic integers, is a Prüfer domain. Just as a Dedekind domain is locally a discrete valuation ring, a Prüfer domain is locally a valuation ring, so that Prüfer domains act as non-noetherian analogues of Dedekind domains. Indeed, a domain that is the direct limit of subrings that are Prüfer domains is a Prüfer domain . Many Prüfe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Module
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as . One often refers to ''the'' trivial object (of a specified category) since every trivial object is isomorphic to any other (under a unique isomorphism). Instances of the zero object include, but are not limited to the following: * As a group, the zero group or trivial group. * As a ring, the zero ring or trivial ring. * As an algebra over a field or algebra over a ring, the trivial algebra. * As a module (over a ring ), the zero module. The term trivial module is also used, although it may be ambiguous, as a ''trivial G-module'' is a G-module with a trivial action. * As a vector space (o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
