Multiplicatively closed setIn 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 geometryThis 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.
