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. Let be a commutative ring and let be the set of elements which are not zero divisors in ; then is a multiplicatively closed set. Hence we may localize the ring at the set to obtain the total quotient ring . If is a domain, then and the total quotient ring is the same as the field of fractions. This justifies the notation , which is sometimes used for the field of fractions as well, since there is no ambiguity in the case of a domain. Since in the construction contains no zero divisors, the natural map is injective, so the total quotient ring is an extension of . For a product ring A × B, the total quotient ring Q(A × B) is the product of total quotient rings Q(A) × Q(B). In particular, if A and B are integral domains, it is the product of quotient fields. For the ring of holomorphic functions on an open set D of complex numbers, the total quotient ring is the ring of meromorphic functions on D, even if D is not connected. In an Artinian ring, all elements are units or zero divisors. Hence the set of non-zero-divisors is the group of units of the ring, , and so . But since all these elements already have inverses, . In a commutative von Neumann regular ring R, the same thing happens. Suppose a in R is not a zero divisor. Then in a von Neumann regular ring a = axa for some x in R, giving the equation a(xa − 1) = 0. Since a is not a zero divisor, xa = 1, showing a is a unit. Here again, . In algebraic geometry one considers a sheaf of total quotient rings on a scheme, and this may be used to give the definition of a Cartier divisor. Let A be a reduced ring that has only finitely many minimal prime ideals, (e.

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Multiplicatively closed set

In abstract algebra, a multiplicatively closed set (or multiplicative set) is a subset S of a ring R such that the following two conditions hold: for all . In other words, S is closed under taking finite products, including the empty product 1. Equivalently, a multiplicative set is a submonoid of the multiplicative monoid of a ring. Multiplicative sets are important especially in commutative algebra, where they are used to build localizations of commutative rings. A subset S of a ring R is called saturated if it is closed under taking divisors: i.

Glossary of algebraic geometry

This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.

Localisation (mathématiques)

En algèbre, la localisation est une des opérations de base de l'algèbre commutative. C'est une méthode qui construit à partir d'un anneau commutatif un nouvel anneau. La construction du corps des fractions est un cas particulier de la localisation. La localisation consiste à rendre inversibles les éléments d'une partie (« partie multiplicative ») de l'anneau. L'exemple le plus connu est le corps des fractions d'un anneau intègre qui se construit en rendant inversibles tous les éléments non nuls de l'anneau.

MATH-215: Rings and fields

C'est un cours introductoire dans la théorie d'anneau et de corps.

Orders That Are Etale-Locally Isomorphic

Eva Bayer Fluckiger

Let R be a semilocal Dedekind domain with fraction field F. It is shown that two hereditary R-orders in central simple F-algebras that become isomorphic after tensoring with F and with some faithfully flat etale R-algebra are isomorphic. On the other hand, ...

AMER MATHEMATICAL SOC2020

Orders that are étale-locally isomorphic

Eva Bayer Fluckiger, Mathieu Huruguen

Let R be a semilocal Dedekind domain with fraction field F. It is shown that two hereditary R-orders in central simple F-algebras that become isomorphic after tensoring with F and with some faithfully flat étale R-algebra are isomorphic. On the other hand, ...

2019

Let R be a semilocal principal ideal domain. Two algebraic objects over R in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all completions of R and its fract ...

Amer Mathematical Soc2017

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