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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'' ('' Satz'' meaning "proposition" or "theorem"). Precisely, if ''R'' is a Noetherian ring and ''I'' is a principal, proper ideal of ''R'', then each minimal prime ideal over ''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 the ... [...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] |
David Rees (mathematician)
David Rees FRS (29 May 1918 – 16 August 2013) was a British professor of pure mathematics at the University of Exeter, having been head of the Mathematics / Mathematical Sciences Department at Exeter from 1958–1983. During the Second World War, Rees was active on Enigma research in Hut 6 at Bletchley Park. Early life Rees was born in Abergavenny to David Rees (1881–), a corn merchant, and his wife Florence Gertrude (Gertie) née Powell (1884–1970), the 4th out of 5 children. Despite periods of ill health and absence, he successfully completed his early education at King Henry VIII Grammar School. Education and career Rees won a scholarship to Sidney Sussex College, Cambridge, supervised by Gordon Welchman and graduating in summer 1939. On completion of his education, he initially worked on semigroup theory; the Rees factor semigroup is named after him. He also characterised completely simple and completely 0-simple semigroups, in what is nowadays known as Rees's t ... [...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] |
Nakayama's Lemma
In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise sense in which finitely generated modules over a commutative ring behave like vector spaces over a field. It is an important tool in algebraic geometry, because it allows local data on algebraic varieties, in the form of modules over local rings, to be studied pointwise as vector spaces over the residue field of the ring. The lemma is named after the Japanese mathematician Tadashi Nakayama and introduced in its present form in , although it was first discovered in the special case of ideals in a commutative ring by Wolfgang Krull and then in general by Goro Azumaya (1951). In the commutative case, the lemma is a simple consequence of a generalized form o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radical Of An Ideal
In ring theory, a branch of mathematics, the radical of an ideal I of a commutative ring is another ideal defined by the property that an element x is in the radical if and only if some power of x is in I. Taking the radical of an ideal is called ''radicalization''. A radical ideal (or semiprime ideal) is an ideal that is equal to its radical. The radical of a primary ideal is a prime ideal. This concept is generalized to non-commutative rings in the Semiprime ring article. Definition The radical of an ideal I in a commutative ring R, denoted by \operatorname(I) or \sqrt, is defined as :\sqrt = \left\, (note that I \subset \sqrt). Intuitively, \sqrt is obtained by taking all roots of elements of I within the ring R. Equivalently, \sqrt is the preimage of the ideal of nilpotent elements (the nilradical) of the quotient ring R/I (via the natural map \pi\colon R\to R/I). The latter proves that \sqrt is an ideal.Here is a direct proof that \sqrt is an ideal. Start with a,b\in\sqrt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symbolic Power (mathematics)
In algebra and algebraic geometry, given a commutative Noetherian ring R and an ideal I in it, the ''n''-th symbolic power of I is the ideal : I^ = \bigcap_ \varphi_P^(I^n R_P) where R_P is the localization of R at P, we set \varphi_P : R \to R_P is the canonical map from a ring to its localization, and the intersection runs through all of the associated primes of R/I. Though this definition does not require I to be prime, this assumption is often worked with because in the case of a prime ideal, the symbolic power can be equivalently defined as the I -primary component of I^n. Very roughly, it consists of functions with zeros of order ''n'' along the variety defined by I. We have: I^ = I and if I is a maximal ideal, then I^ = I^n. Symbolic powers induce the following chain of ideals: : I^=R\supset I=I^\supset I^\supset I^\supset I^\supset \cdots Uses The study and use of symbolic powers has a long history in commutative algebra. Krull’s famous proof of his principa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Ideal
In mathematics, specifically commutative algebra, a proper ideal ''Q'' of a commutative ring ''A'' is said to be primary if whenever ''xy'' is an element of ''Q'' then ''x'' or ''y''''n'' is also an element of ''Q'', for some ''n'' > 0. For example, in the ring of integers Z, (''p''''n'') is a primary ideal if ''p'' is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem. Consequently, an irreducible ideal of a Noetherian ring is primary. Various methods of generalizing primary ideals to noncommutative rings exist, but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity. Examples and properties * The definition can be rephrased in a more symmetric manner: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization (commutative Algebra)
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term ''localization'' originated in algebraic geometry: if ''R'' is a ring of functions defined on some geometric object (algebraic variety) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localizes ''R'' wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Éléments De Mathématique
''Éléments de mathématique'' (English: ''Elements of Mathematics'') is a series of mathematics books written by the pseudonymous French collective Nicolas Bourbaki. Begun in 1939, the series has been published in several volumes, and remains in progress. The series is noted as a large-scale, self-contained, formal treatment of mathematics. The members of the Bourbaki group originally intended the work as a textbook on analysis, with the working title ''Traité d'analyse'' (''Treatise on Analysis''). While planning the structure of the work they became more ambitious, expanding its scope to cover several branches of modern mathematics. Once the plan of the work was expanded to treat other fields in depth, the title ''Éléments de mathématique'' was adopted. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups and Lie algebras. The unusual singular "mathématique" (mathematic) of the title is deliberate, meant to convey the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject. Krull was born and went to school in Baden-Baden. He attended the Universities of Freiburg, Rostock and finally Göttingen from 1919–1921, where he earned his doctorate under Alfred Loewy. He worked as an instructor and professor at Freiburg, then spent a decade at the University of Erlangen. In 1939 Krull moved to become chair at the University of Bonn, where he remained for the rest of his life. Wolfgang Krull was a member of the Nazi Party. His 35 doctoral students include Wilfried Brauer, Karl-Otto Stöhr and Jürgen Neukirch. See also * Cohen structure theorem * Jacobson ring * Local ring * Prime ideal * Real algebraic geometry * Regular local ring * Valuation ring * Krull dimension * Krull ring * Krull topology * Krull–Azumaya theorem * Krull–Schmidt category * Krull–S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Theorem Of Dimension Theory (algebra)
In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme). The need of a ''theory'' for such an apparently simple notion results from the existence of many definitions of the dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety). A large part of dimension theory consists in studying the conditions under which several dimensions are equal, and many important classes of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological dimension is equal to the Krull dimension. The theory is simpler for commutative rings that are finitely generated algebras over a field, which are also quotient rings of polynomial rings in a finite number of indeterminates over a field. In this case, which is the algebraic counterpart of the case of affine algebr ... [...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 of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
