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Stable Range (topology)
In mathematics, particular in abstract algebra and algebraic K-theory, the stable range of a ring R is the smallest integer n such that whenever v_0,v_1,...,v_n in R generate the unit ideal (they form a unimodular row), there exist some t_1,...,t_nin R such that the elements v_i - v_0t_i for 1\le i \le n also generate the unit ideal. If R is a commutative Noetherian ring of Krull dimension d , then the stable range of R is at most d+1 (a theorem of Bass). Bass stable range The Bass stable range condition SR_m refers to precisely the same notion, but for historical reasons it is indexed differently: a ring R satisfiesSR_m if for any v_1,...,v_m in R generating the unit ideal there exist t_2,...,t_m in R such that v_i - v_1t_i for 2\le i \le m generate the unit ideal. Comparing with the above definition, a ring with stable range n satisfies SR_. In particular, Bass's theorem states that a commutative Noetherian ring of Krull dimension In commutative algebra, the Krull dimensio ... [...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 poin ... [...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, alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the ''K''-groups of the integers. ''K''-theory was discovered in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only ''K''0, the zeroth ''K''-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic ''K''-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the resu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unimodular Row
This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of module theory. In this article, all rings are assumed to be commutative with identity 1. !$@ A B C D E F G H . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull Dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal ''I'' in a polynomial ring ''R'' is the Krull dimension of ''R''/''I''. A field ''k'' has Krull dimension 0; more generally, ''k'' 'x''1, ..., ''x''''n''has Krull dimension ''n''. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent. There are several other ways that have been used to define the dimension of a ring. Most of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ideal (ring Theory)
In ring theory, a branch of abstract algebra, 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 elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Weibel
Charles Alexander Weibel (born October 28, 1950 in Terre Haute, Indiana) is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra. Weibel studied physics and mathematics at the University of Michigan, earning bachelor's degrees in both subjects in 1972. He was awarded a master's degree by the University of Chicago in 1973 and achieved his doctorate in 1977 under the supervision of Richard Swan (''Homotopy in Algebraic K-Theory''). From 1970 to 1976 he was an "Operations Research Analyst" at Standard Oil of Indiana, and from 1977 to 1978 was at the Institute for Advanced Study. In 1978 he became an assistant professor at the University of Pennsylvania. In 1980 he became an assistant professor at Rutgers University, where he was promoted to professor in 1989. He joined Vladimir Voevodsky and Markus Rost in proving the ( motivic) Bloch–Kato conjecture (2009). It is a generalization of the Milnor conjecture of algebraic K-theory, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
