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Mori–Nagata Theorem
In algebra, the Mori–Nagata theorem introduced by and , states the following: let ''A'' be a noetherian reduced commutative ring with the total ring of fractions ''K''. Then the integral closure of ''A'' in ''K'' is a direct product of ''r'' Krull domains, where ''r'' is the number of minimal prime ideals of ''A''. The theorem is a partial generalization of the Krull–Akizuki theorem, which concerns a one-dimensional noetherian domain. A consequence of the theorem is that if ''R'' is a Nagata ring In commutative algebra, an N-1 ring is an integral domain A whose integral closure in its quotient field is a finitely generated A-module. It is called a Japanese ring (or an N-2 ring) if for every finite extension L of its quotient field K, the ..., then every ''R''-subalgebra of finite type is again a Nagata ring . The Mori–Nagata theorem follows from Matijevic's theorem. References * * * * Commutative algebra Theorems in ring theory {{abstract-algebra-stub ... [...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] |
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] |
Commutative Ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot ... [...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 divisor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Closure
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a''''j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is a root of a monic polynomial over ''A''. The set of elements of ''B'' that are integral over ''A'' is called the integral closure of ''A'' in ''B''. It is a subring of ''B'' containing ''A''. If every element of ''B'' is integral over ''A'', then we say that ''B'' is integral over ''A'', or equivalently ''B'' is an integral extension of ''A''. If ''A'', ''B'' are fields, then the notions of "integral over" and of an "integral extension" are precisely " algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial). The case of greatest interest in number theory is that of complex numbers integral over Z (e.g., \sqrt or 1+i); in this context, the integral elements are usual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the product in category theory, which formalizes these notions. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept. Examples * If we think of \R as the set of real numbers, then the direct product \R \times \R is just the Cartesian product \. * If we think of \R as the group of real numbers under addition, then the direct product \R\times \R still has \ as its underlying set. The difference between this and the preceding example is that \R \times \R is now a group, and so we have to also say how to add their ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull Domain
In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1. In this article, a ring is commutative and has unity. Formal definition Let A be an integral domain and let P be the set of all prime ideals of A of height one, that is, the set of all prime ideals properly containing no nonzero prime ideal. Then A is a Krull ring if # A_ is a discrete valuation ring for all \mathfrak \in P , # A is the intersection of these discrete valuation rings (considered as subrings of the quotient field of A ). #Any nonzero element of A is contained in only a finite number of height 1 prime ideals. It is also possible to characterize Krull rings by mean of valuations only: An integral domain A is a Krull ring if there exists a family \ _ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ideals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Krull–Akizuki Theorem
In algebra, the Krull–Akizuki theorem states the following: let ''A'' be a one-dimensional reduced noetherian ring, ''K'' its total ring of fractions. If ''B'' is a subring of a finite extension ''L'' of ''K'' containing ''A'' then ''B'' is a one-dimensional noetherian ring. Furthermore, for every nonzero ideal ''I'' of ''B'', B/I is finite over ''A''. Note that the theorem does not say that ''B'' is finite over ''A''. The theorem does not extend to higher dimension. One important consequence of the theorem is that the integral closure of a Dedekind domain ''A'' in a finite extension of the field of fractions of ''A'' is again a Dedekind domain. This consequence does generalize to a higher dimension: the Mori–Nagata theorem states that the integral closure of a noetherian domain is a Krull domain In commutative algebra, a Krull ring, or Krull domain, is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull in 1931. They ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nagata Ring
In commutative algebra, an N-1 ring is an integral domain A whose integral closure in its quotient field is a finitely generated A-module. It is called a Japanese ring (or an N-2 ring) if for every finite extension L of its quotient field K, the integral closure of A in L is a finitely generated A-module (or equivalently a finite A-algebra). A ring is called universally Japanese if every finitely generated integral domain over it is Japanese, and is called a Nagata ring, named for Masayoshi Nagata, or a pseudo-geometric ring if it is Noetherian and universally Japanese (or, which turns out to be the same, if it is Noetherian and all of its quotients by a prime ideal are N-2 rings). A ring is called geometric if it is the local ring of an algebraic variety or a completion of such a local ring , but this concept is not used much. Examples Fields and rings of polynomials or power series in finitely many indeterminates over fields are examples of Japanese rings. Another important exa ... [...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] |
