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Catenary Ring
In mathematics, a commutative ring ''R'' is catenary if for any pair of prime ideals ''p'', ''q'', any two strictly increasing chains :''p'' = ''p''0 ⊂ ''p''1 ⊂ ... ⊂ ''p''''n'' = ''q'' of prime ideals are contained in maximal strictly increasing chains from ''p'' to ''q'' of the same (finite) length. In a geometric situation, in which the dimension of an algebraic variety attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain ''n'' is usually the difference in dimensions. A ring is called universally catenary if all finitely generated algebras over it are catenary rings. The word 'catenary' is derived from the Latin word ''catena'', which means "chain". There is the following chain of inclusions. Dimension formula Suppose that ''A'' is a Noetherian domain and ''B'' is a domain containing ''A'' that is finitely generated over ''A''. If ''P'' is a prime ideal of ''B'' and ''p'' its intersection with ''A'', then :\te ... [...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] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form \sum_^\infty a_nx^n=a_0+a_1x+ a_2x^2+\cdots, where the a_n, called ''coefficients'', are numbers or, more generally, elements of some ring, and the x^n are formal powers of the symbol x that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gorenstein Local Ring
In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring ''R'' with finite injective dimension as an ''R''-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring is self-dual in some sense. Gorenstein rings were introduced by Grothendieck in his 1961 seminar (published in ). The name comes from a duality property of singular plane curves studied by (who was fond of claiming that he did not understand the definition of a Gorenstein ring ). The zero-dimensional case had been studied by . and publicized the concept of Gorenstein rings. Frobenius rings are noncommutative analogs of zero-dimensional Gorenstein rings. Gorenstein schemes are the geometric version of Gorenstein rings. For Noetherian local rings, there is the following chain of inclusions. Definitions A Gorenstein ring is a commutative Noetherian ring such that each localization at a prime ideal is a Gorenstein local ring, as de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of subclass (set theory), class inclusions: Definition and first examples Definition A ''ring'' is a Set (mathematics), set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Ring
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. The concept of local rings was introduced by Wolfgang Krull in 1938 under the name ''Stellenringe''. The English term ''local ring'' is due to Zariski. Definition and first consequences A ring ''R'' is a local ring if it has any one of the following equivalent properties: * ''R'' has a unique maximal left ideal. * ''R'' has a unique maximal right ideal. * 1 ≠ 0 and the sum of any two non- units in ''R'' is a non-unit. * 1 ≠ 0 and if ''x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formally Catenary Ring
In algebra, specifically in the theory of commutative rings, a quasi-unmixed ring (also called a formally equidimensional ring in EGA) is a Noetherian ring A such that for each prime ideal ''p'', the completion of the localization ''Ap'' is equidimensional, i.e. for each minimal prime ideal ''q'' in the completion \widehat, \dim \widehat/q = \dim A_p = the Krull dimension of ''Ap''. Equivalent conditions A Noetherian integral domain is quasi-unmixed if and only if it satisfies Nagata's altitude formula. (See also: #formally catenary ring below.) Precisely, a quasi-unmixed ring is a ring in which the unmixed theorem, which characterizes a Cohen–Macaulay ring, holds for integral closure of an ideal; specifically, for a Noetherian ring A, the following are equivalent: *A is quasi-unmixed. *For each ideal ''I'' generated by a number of elements equal to its height, the integral closure \overline is unmixed in height (each prime divisor has the same height as the others). *For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excellent Ring
In commutative algebra, a quasi-excellent ring is a Noetherian ring, Noetherian commutative ring that behaves well with respect to the operation of completion of a ring, completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" ring (mathematics), rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-excellent Ring
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved" rings containing most of the rings that occur in number theory and algebraic geometry. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but Masayoshi Nagata and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined by Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are conjectured to be the base rings for which the problem of resolution of singularities can be solved; showed this in characteristic 0, but th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobson Radical
In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R- modules. It happens that substituting "left" in place of "right" in the definition yields the same ideal, and so the notion is left–right symmetric. The Jacobson radical of a ring is frequently denoted by J(R) or \operatorname(R); the former notation will be preferred in this article to avoid confusion with other radicals of a ring. The Jacobson radical is named after Nathan Jacobson, who was the first to study it for arbitrary rings in . The Jacobson radical of a ring has numerous internal characterizations, including a few definitions that successfully extend the notion to non-unital rings. The radical of a module extends the definition of the Jacobson radical to include modules. The Jacobson radical plays a prominent role in many ring- and module-theoretic results, such as Nakayama's lemma. Definitions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A Ring
Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is affected by touch or other sensation; see Allochiria * Neurologic localization, in neurology, the process of deducing the location of injury based on symptoms and neurological examination * Nuclear localization signal, an amino acid sequence on the surface of a protein which acts like a 'tag' to localize the protein in the cell * Sound localization, a listener's ability to identify the location or origin of a detected sound * Subcellular localization, organization of cellular components into different regions of a cell Engineering and technology * GSM localization, determining the location of an active cell phone or wireless transceiver * Robot localization, figuring out robot's position in an environment * Indoor positioning system, a networ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of subclass (set theory), class inclusions: Definition and first examples Definition A ''ring'' is a Set (mathematics), set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
