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.e., whenever a product xy is in S, the elements x and y are in S too. Examples of multiplicative sets include: the set-theoretic complement of a prime ideal in a commutative ring; the set {1, x, x2, x3, ...}, where x is an element of a ring; the set of units of a ring; the set of non-zero-divisors in a ring; 1 + I for an ideal I. the Jordan–Pólya numbers, the multiplicative closure of the factorials An ideal P of a commutative ring R is prime if and only if its complement R \ P is multiplicatively closed. A subset S is both saturated and multiplicatively closed if and only if S is the complement of a union of prime ideals. In particular, the complement of a prime ideal is both saturated and multiplicatively closed. The intersection of a family of multiplicative sets is a multiplicative set. The intersection of a family of saturated sets is saturated.

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Concepts associés (15)

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Noncommutative ring

In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings. Sometimes the term noncommutative ring is used instead of ring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative.

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.

Clôture (mathématiques)

On parle de clôture ou de fermeture en mathématiques dans des contextes très divers. Quelques exemples sont listés ci-dessous. En mathématiques, on dit qu'une partie A d'un ensemble E est stable (ou close) pour une opération définie sur E si cette opération, appliquée à des éléments de A, produit toujours un élément de A. Par exemple, l'ensemble des nombres réels est stable par soustraction, tandis que l'ensemble des entiers naturels ne l'est pas (la différence de deux entiers naturels est parfois un entier relatif strictement négatif).

Revisiting the nilpotent polynomial Hales–Jewett theorem

Florian Karl Richter

Answering a question posed by Bergelson and Leibman in [6], we establish a nilpotent version of the Polynomial Hales–Jewett Theorem that contains the main theorem in [6] as a special case. Important to the formulation and the proof of our main theorem is t ...

2017

Localisation en algèbre MATH-311: Algebra IV - rings and modules

Couvre le concept de localisation en algèbre, en se concentrant sur la réalisation d'un anneau multiplicatif fermé.

Équation de Fermat : Solutions intégrales et entiers gaussiens

Discute de l'équation de Fermat et des propriétés des entiers gaussiens.

Anneaux et modules : Extensions intégrales MATH-311: Algebra IV - rings and modules

Couvre les extensions intégrales dans les anneaux et les modules, en discutant des propriétés et en fournissant des exemples illustratifs.