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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 ... [...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] |
Maximal Ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals contained between ''I'' and ''R''. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields. The set of maximal ideals of a unital commutative ring ''R'', typically equipped with the Zariski topology, is known as the maximal spectrum of ''R'' and is variously denoted m-Spec ''R'', Specm ''R'', MaxSpec ''R'', or Spm ''R''. In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. Since a one-sided maximal ideal ''A'' is not necessarily ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum Of A Ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with a sheaf of rings. Zariski topology For any ideal I of R, define V_I to be the set of prime ideals containing I. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\big\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows: For f\in R, define D_f to be the set of prime ideals of R not containing f. Then each D_f is an open subset of \operatorname(R), and \big\ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: In fact, the maximal ideals in R are precisely the closed points in this topology. By the same reasoning, \operatorname(R) is not, in general, a T1 space. However, \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Definition Codimension is a ''relative'' concept: it is only defined for one object ''inside'' another. There is no “codimension of a vector space (in isolation)”, only the codimension of a vector ''sub''space. If ''W'' is a linear subspace of a finite-dimensional vector space ''V'', then the codimension of ''W'' in ''V'' is the difference between the dimensions: :\operatorname(W) = \dim(V) - \dim(W). It is the complement of the dimension of ''W,'' in that, with the dimension of ''W,'' it adds up to the dimension of the ambient space ''V:'' :\dim(W) + \operatorname(W) = \dim(V). Simi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descending Chain 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. These conditions played an important role in the development of the structure theory of commutative rings in the works of David Hilbert, Emmy Noether, 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 strictly ascending sequence : a_1 < a_2 < a_3 < \cdots of elements of ''P'' exists. Equivalently, every weakly ascending sequence : of elements of ''P'' eventually stabilizes, meaning that there exists a positive integer ''n'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull's Principal Ideal Theorem
In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull (1899–1971), gives a bound on the height of a principal ideal in a commutative Noetherian ring. The theorem is sometimes referred to by its German name, ''Krulls Hauptidealsatz'' (from ' ("Principal") + ' + ' ("theorem")). Precisely, if ''R'' is a Noetherian ring and ''I'' is a principal, proper ideal of ''R'', then each minimal prime ideal containing ''I'' has height at most one. This theorem can be generalized to ideals that are not principal, and the result is often called Krull's height theorem. This says that if ''R'' is a Noetherian ring and ''I'' is a proper ideal generated by ''n'' elements of ''R'', then each minimal prime over ''I'' has height at most ''n''. The converse is also true: if a prime ideal has height ''n'', then it is a minimal prime ideal over an ideal generated by ''n'' elements. The principal ideal theorem and the generalization, the height theorem, both follow from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universally Catenary
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] |
Masayoshi Nagata
Masayoshi Nagata ( Japanese: 永田 雅宜 ''Nagata Masayoshi''; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra. Work Nagata's compactification theorem shows that algebraic varieties can be embedded in complete varieties. The Chevalley–Iwahori–Nagata theorem describes the quotient of a variety by a group. In 1959, he introduced a counterexample to the general case of Hilbert's fourteenth problem on invariant theory. His 1962 book on local rings contains several other counterexamples he found, such as a commutative Noetherian ring that is not catenary, and a commutative Noetherian ring of infinite dimension. Nagata's conjecture on curves concerns the minimum degree of a plane curve specified to have given multiplicities at given points; see also Seshadri constant. Nagata's conjecture on automorphisms concerns the existence of wild automorphisms of polynomial algebras in three variables. R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
