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Serre's Criterion
In algebra, Serre's criterion for normality, introduced by Jean-Pierre Serre, gives necessary and sufficient conditions for a commutative Noetherian ring ''A'' to be a normal ring. The criterion involves the following two conditions for ''A'': *R_k: A_ is a regular local ring for any prime ideal \mathfrak of height ≤ ''k''. *S_k: \operatorname A_ \ge \inf \ for any prime ideal \mathfrak. The statement is: *''A'' is a reduced ring \Leftrightarrow R_0, S_1 hold. *''A'' is a normal ring \Leftrightarrow R_1, S_2 hold. *''A'' is a Cohen–Macaulay ring \Leftrightarrow S_k hold for all ''k''. Items 1, 3 trivially follow from the definitions. Item 2 is much deeper. For an integral domain, the criterion is due to Krull. The general case is due to Serre. Proof Sufficiency (After EGA IV. Theorem 5.8.6.) Suppose ''A'' satisfies ''S''2 and ''R''1. Then ''A'' in particular satisfies ''S''1 and ''R''0; hence, it is reduced. If \mathfrak_i, \, 1 \le i \le r are the minimal prime ideals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Biography Personal life Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and then from 1945 to 1948 at the École Normale Supérieure in Paris. He was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris. In 1956 he was elected professor at the Collège de France, a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter is the former French diplomat, historian and writer Claudine Monteil. The French mathematician Denis S ... [...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 Laskerâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Ring
In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root of a monic polynomial with coefficients in ''A,'' then ''x'' is itself an element of ''A.'' Many well-studied domains are integrally closed: fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed. Note that integrally closed domains appear in the following chain of class inclusions: Basic properties Let ''A'' be an integrally closed domain with field of fractions ''K'' and let ''L'' be a field extension of ''K''. Then ''x''∈''L'' is integral over ''A'' if and only if it is algebraic over ''K'' and its minimal polynomial over ''K'' has coefficients in ''A''. In particular, this means that any element of ''L'' integral over ''A'' is root of a monic polynomial in ''A'' 'X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Local Ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ideal m, and suppose ''a''1, ..., ''a''''n'' is a minimal set of generators of m. Then by Krull's principal ideal theorem ''n'' ≥ dim ''A'', and ''A'' is defined to be regular if ''n'' = dim ''A''. The appellation ''regular'' is justified by the geometric meaning. A point ''x'' on an algebraic variety ''X'' is nonsingular if and only if the local ring \mathcal_ of germs at ''x'' is regular. (See also: regular scheme.) Regular local rings are ''not'' related to von Neumann regular rings. For Noetherian local rings, there is the following chain of inclusions: Characterizations There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if A is a Noetherian local ring with maximal idea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Ring
In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' = 0. A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring ''R'' form an ideal of ''R'', called the nilradical of ''R''; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring ''R/I'' is reduced if and only if ''I'' is a radical ideal. Let ''D'' be the set of all zero-divisors in a reduced ring ''R''. Then ''D'' is the union of all minimal prime ideals. Over a Noetherian ring ''R'', we say a finitely generated module ''M'' has locally constant rank if \mathfrak \mapsto \operatorname_ ... [...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 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 local ring ''R'', a finite (i.e. finitely generated) ''R''-module M\neq 0 is a ''Cohen-Macaulay module'' if \mathrm(M) = \mathrm(M) (in general we have: \mathrm(M) \leq \mathrm(M), see Auslander–Buchsbaum formula for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Ring Of Fractions
In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embeds ''R'' in a larger ring, giving every non-zero-divisor of ''R'' an inverse in the larger ring. If the homomorphism from ''R'' to the new ring is to be injective, no further elements can be given an inverse. Definition Let R be a commutative ring and let S be the set of elements which are not zero divisors in R; then S is a multiplicatively closed set. Hence we may localize the ring R at the set S to obtain the total quotient ring S^R=Q(R). If R is a domain, then S=R-\ and the total quotient ring is the same as the field of fractions. This justifies the notation Q(R), which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain. Since S in the construction contains no zero diviso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unmixed Theorem
''Unmixed'' is the second album by house production duo Freemasons. It was released on 29 October 2007, with the lead single "Uninvited" being released one week earlier. The album is unique, as unlike most DJ/Producers, the tracks are all unmixed full length versions. In addition, producer samples packs created by the Freemasons were included on a data portion of the album. Track listing Samples, covers and interpolations * "Uninvited" is a cover of Alanis Morissette's song. * "If" is a cover of Jackie Moore's song, released in 1973 on her album ''Sweet Charlie Babe''. * "Nothing But A Heartache" is a cover of The Flirtations' song. * "Love on My Mind" incorporates lyrics from Tina Turner's song "When the Heartache Is Over" and samples Jackie Moore's song "This Time Baby". * "Watchin'" incorporates elements from Deborah Cox's song "It's Over Now", released in 1998 on her album '' One Wish''. * "Love Don't Live Here Anymore" is a cover of Rose Royce's song. * "Pacific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associated Prime
In abstract algebra, an associated prime of a module ''M'' over a ring ''R'' is a type of prime ideal of ''R'' that arises as an annihilator of a (prime) submodule of ''M''. The set of associated primes is usually denoted by \operatorname_R(M), and sometimes called the ''assassin'' or ''assassinator'' of (word play between the notation and the fact that an associated prime is an ''annihilator''). In commutative algebra, associated primes are linked to the Lasker–Noether primary decomposition of ideals in commutative Noetherian rings. Specifically, if an ideal ''J'' is decomposed as a finite intersection of primary ideals, the radicals of these primary ideals are prime ideals, and this set of prime ideals coincides with \operatorname_R(R/J). Also linked with the concept of "associated primes" of the ideal are the notions of isolated primes and embedded primes. Definitions A nonzero ''R'' module ''N'' is called a prime module if the annihilator \mathrm_R(N)=\mathrm_R(N')\, f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primary Decomposition
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 submodules ... [...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] |
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] |
