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Goldie Ring
In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring ''R'' that has finite uniform dimension (="finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of ''R''. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on ''all'' right ideals. This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain. A con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ore Domain
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The ''right Ore condition'' for a multiplicative subset ''S'' of a ring ''R'' is that for and , the intersection . A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly. General idea The goal is to construct the right ring of fractions ''R'' 'S''−1with respect to a multiplicative subset ''S''. In other words, we want to work with elements of the form ''as''−1 and have a ring structure on the set ''R'' 'S''−1 The problem is that there is no obvious interpretation of the product (''as''−1)(''bt''−1); indeed, we need a method to "move" ''s''−1 past ''b''. This means that we need to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ore Ring
In mathematics, especially in the area of algebra known as ring theory, the Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring. The ''right Ore condition'' for a multiplicative subset ''S'' of a ring ''R'' is that for and , the intersection . A (non-commutative) domain for which the set of non-zero elements satisfies the right Ore condition is called a right Ore domain. The left case is defined similarly. General idea The goal is to construct the right ring of fractions ''R'' 'S''−1with respect to a multiplicative subset ''S''. In other words, we want to work with elements of the form ''as''−1 and have a ring structure on the set ''R'' 'S''−1 The problem is that there is no obvious interpretation of the product (''as''−1)(''bt''−1); indeed, we need a method to "move" ''s''−1 past ''b''. This means that we need to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Condition
In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some algebraic structures, most importantly ideals in certain commutative rings.Jacobson (2009), p. 142 and 147 These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noethe ..., and Emil Artin. The conditions themselves can be stated in an abstract form, so that they make sense for any partially ordered set. This point of view is useful in abstract algebraic dimension theory due to Gabriel and Rentschler. Definition A partially ordered set (poset) ''P'' is said to satisfy the ascending chain condition (ACC) if no infinite s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonsingular Ring
In the branches of abstract algebra known as ring theory and module theory, each right (resp. left) ''R''-module ''M'' has a singular submodule consisting of elements whose annihilators are essential right (resp. left) ideals in ''R''. In set notation it is usually denoted as \mathcal(M)=\\,. For general rings, \mathcal(M) is a good generalization of the torsion submodule tors(''M'') which is most often defined for domains. In the case that ''R'' is a commutative domain, \operatorname(M) = \mathcal(M). If ''R'' is any ring, \mathcal(R_R) is defined considering ''R'' as a right module, and in this case \mathcal(R_R) is a two-sided ideal of ''R'' called the right singular ideal of ''R''. The left handed analogue \mathcal(_R R) is defined similarly. It is possible for \mathcal(R_R) \neq \mathcal(_R R). Definitions Here are several definitions used when studying singular submodules and singular ideals. In the following, ''M'' is an ''R''-module: *''M'' is called a singular modul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nil Ideal
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent., p. 194 The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil. Unfortunately the set of nil elements does not always form an ideal for noncommutative rings. Nil ideals are still associated with interesting open questions, especially the unsolved Köthe conjecture. Commutative rings In commutative rings, the nil ideals are better understood than in noncommutative rings, primarily because in commutative rings, products involving nilpotent elements and sums of nilpotent elements are both nilpotent. This is because if ''a'' and ''b'' are nilpotent elements of ''R'' with ''a''n=0 and ''b''m=0, and r is any element of R, then (''a''·''r'')n = ''a''n·''r''n = 0, and by the binomial theorem, (''a''+''b'')m+n=0. Therefore, the set of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Element (ring Theory)
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). If the ring is commutative, then the left and right zero divisors are the same. An element of a ring that is not a left zero divisor is called left regular or left cancellable. Similarly, an element of a ring that is not a right zero divisor is called right regular or right cancellable. An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essential Submodule
In mathematics, specifically module theory, given a ring ''R'' and an ''R''-module ''M'' with a submodule ''N'', the module ''M'' is said to be an essential extension of ''N'' (or ''N'' is said to be an essential submodule or large submodule of ''M'') if for every submodule ''H'' of ''M'', :H\cap N=\\, implies that H=\\, As a special case, an essential left ideal of ''R'' is a left ideal that is essential as a submodule of the left module ''R''''R''. The left ideal has non-zero intersection with any non-zero left ideal of ''R''. Analogously, an essential right ideal is exactly an essential submodule of the right ''R'' module ''R''''R''. The usual notations for essential extensions include the following two expressions: :N\subseteq_e M\, , and N\trianglelefteq M The dual notion of an essential submodule is that of superfluous submodule (or small submodule). A submodule ''N'' is superfluous if for any other submodule ''H'', :N+H=M\, implies that H=M\,. The usual notations for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''Undergraduate algebra'', Springer, 2005; V.§3. (alternative notations: Mat''n''(''R'') and ). Some sets of infinite matrices form infinite matrix rings. Any subring of a matrix ring is a matrix ring. Over a rng, one can form matrix rngs. When ''R'' is a commutative ring, the matrix ring M''n''(''R'') is an associative algebra over ''R'', and may be called a matrix algebra. In this setting, if ''M'' is a matrix and ''r'' is in ''R'', then the matrix ''rM'' is the matrix ''M'' with each of its entries multiplied by ''r''. Examples * The set of all matrices over ''R'', denoted M''n''(''R''). This is sometimes called the "full ring of ''n''-by-''n'' matrices". * The set of all upper triangular matrices over ''R''. * The set of all l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Ring
In abstract algebra, a nonzero ring ''R'' is a prime ring if for any two elements ''a'' and ''b'' of ''R'', ''arb'' = 0 for all ''r'' in ''R'' implies that either ''a'' = 0 or ''b'' = 0. This definition can be regarded as a simultaneous generalization of both integral domains and simple rings. Although this article discusses the above definition, prime ring may also refer to the minimal non-zero subring of a field, which is generated by its identity element 1, and determined by its characteristic. For a characteristic 0 field, the prime ring is the integers, and for a characteristic ''p'' field (with ''p'' a prime number) the prime ring is the finite field of order ''p'' (cf. Prime field).Page 90 of Equivalent definitions A ring ''R'' is prime if and only if the zero ideal is a prime ideal in the noncommutative sense. This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for ''R'' to be a prime ring: *For any two ideals ''A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Right Ideal Ring
In mathematics, a principal right (left) ideal ring is a ring ''R'' in which every right (left) ideal is of the form ''xR'' (''Rx'') for some element ''x'' of ''R''. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when ''R'' is a commutative ring, ''R'' can be called a principal ideal ring, or simply principal ring. If only the finitely generated right ideals of ''R'' are principal, then ''R'' is called a right Bézout ring. Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains. A commutative principal ideal ring which is also an integral domain is said to be a ''principal ideal domain'' (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain. General properties If ''R'' is a principal right ideal ring, then it is certainly a right Noetherian rin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Domain
In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element ''a'' has the cancellation property, that is, if , an equality implies . "Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings. Some sources, notably Lang, use the term entir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
