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Minimal Prime Ideal
In mathematics, especially in commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes. Definition A prime ideal ''P'' is said to be a minimal prime ideal over an ideal ''I'' if it is minimal among all prime ideals containing ''I''. (Note: if ''I'' is a prime ideal, then ''I'' is the only minimal prime over it.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal. A minimal prime ideal over an ideal ''I'' in a Noetherian ring ''R'' is precisely a minimal associated prime (also called isolated prime) of R/I; this follows for instance from the primary decomposition of ''I''. Examples * In a commutative artinian ring, every maximal ideal is a minimal prime ideal. * In an integral domain, the only minimal prime ideal is the zero ideal. * In the ring Z of integers, the minimal prime ide ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symbolic Power Of An Ideal
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 princip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohen–Macaulay Ring
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebraic geometry, algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central role in commutative algebra: they form a very broad class, and yet they are well understood in many ways. They are named for , who proved the unmixedness theorem for polynomial rings, and for , who proved the unmixedness theorem for formal power series rings. All Cohen–Macaulay rings have the unmixedness property. For Noetherian local rings, there is the following chain of inclusions. Definition For a commutative noetherian ring, Noetherian local ring ''R'', a finite (i.e. Finitely generated module, finitely generated) ''R''-module M\neq 0 is a ''Cohen-Macaulay module'' if \mathrm(M) = \mathrm(M) (in general we have: \mathrm ... [...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 them coinci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Component
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by . The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artinian Ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition. Precisely, a ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other. The Artin–Wedderburn theorem char ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Divisor
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 re ... [...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\sqr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ascending Chain Conditions On Radical Ideals
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an such that: I_=I_=\cdots. Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Las ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Noether'' (1907/08, NR. 2988); reproduced in: ''Emmy Noether, Gesammelte Abhandlungen – Collected Papers,'' ed. N. Jacobson 1983; online facsimile aphysikerinnen.de/noetherlebenslauf.html). Sometimes ''Emmy'' is mistakenly reported as a short form for ''Amalie'', or misreported as "Emily". e.g. (, ; ; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She discovered Noether's First and Second Theorem, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
