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: 1 ∈ S...

commonly done with respect to a multiplicatively closed set S (also called a multiplicative set or a multiplicative system) of elements of a ring R,...

whole ring.) An ideal I is prime if and only if its set-theoretic complement is multiplicatively closed. Every nonzero ring contains at least one prime ideal...

{\displaystyle S} be the set of elements that are not zero divisors in R {\displaystyle R} ; then S {\displaystyle S} is a multiplicatively closed set. Hence we may...

follows that Tor commutes with localization. That is, for a multiplicatively closed set S in R, S − 1 Tor i R ⁡ ( A , B ) ≅ Tor i S − 1 R ⁡ ( S − 1 A...

Ext commutes with localization, in the sense that for every multiplicatively closed set S in R, every R-module B, and every integer i, S − 1 Ext R i...

Saturated measure, if every locally measurable set is also measurable Saturated multiplicatively closed sets, a concept in ring theory "Saturation (song)"...

is a Krull domain. If S ⊂ A {\displaystyle S\subset A} is a multiplicatively closed set not containing 0, the ring of quotients S − 1 A {\displaystyle...

mathematics, an expression or equation is in closed form if it is formed with constants, variables and a finite set of basic functions connected by arithmetic...

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces...

zero vector 0 in span S, that span S is closed under addition, and that span S is closed under scalar multiplication. Letting S = { v 1 , v 2 , … , v n }...

256, ... (sequence A001013 in the OEIS) They form the smallest multiplicatively closed set containing all of the factorials. The n {\displaystyle n} th...

commutative). More generally, let M be a module over a ring R and S be a multiplicatively closed subset of R. An element m of M is called an S-torsion element if...

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on...

calculations, involving these operations and relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join...

evaluated under the ordinary multiplication of natural numbers. The smallest algebraically closed field of nimbers is the set of nimbers less than the ordinal...

localization are the following: when R is a commutative ring and S a multiplicatively closed subset p ↦ p [ S − 1 ] {\displaystyle {\mathfrak {p}}\mapsto {\mathfrak...

natural numbers, Z {\displaystyle \mathbb {Z} } is closed under the operations of addition and multiplication, that is, the sum and product of any two integers...

ISBN 978-1-4832-8079-0. ...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication Davisson, Schuyler Colfax...

set (the empty product) should be considered to be one, since one is the identity element for multiplication. A derangement is a permutation of a set...

is closed in X . {\displaystyle X.} The sum of a compact set and a closed set is closed. However, the sum of two closed subsets may fail to be closed (see...

other sets. A set may have a finite number of elements or be an infinite set. There is a unique set with no elements, called the empty set; a set with...

sorts of cones—is a subset of a vector space that is closed under positive scalar multiplication; that is, C is a cone if x ∈ C {\displaystyle x\in C}...

written multiplicatively, denoted by juxtaposition. Then H is a subgroup of G if and only if H is nonempty and closed under products and inverses. Closed under...

Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic...

algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying...

needed] and addition and multiplication are often added as axioms. The respective functions and relations are constructed in set theory or second-order...

language of real closed fields L rcf {\displaystyle {\mathcal {L}}_{\text{rcf}}} includes symbols for the operations of addition and multiplication, the constants...

mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation...

rendered invertible, i.e. multiplicative inverses are added to the ring. Concretely, if S {\displaystyle S} is a multiplicatively closed subset of R {\displaystyle...