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,...

{\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...

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...

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

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...

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)"...

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

function is multiplicative precisely when C {\displaystyle C} is closed under multiplication of coprime elements. There are also other sets (not closed under...

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...

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 }...

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

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...

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

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...

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

denominator set if it satisfies the following three conditions for every a, b in R, and s, t in S: st in S; (The set S is multiplicatively closed.) aS ∩ sR...

indecomposable. The finite ordinal 2 is multiplicatively indecomposable since 1·1 = 1 < 2. Besides 2, the multiplicatively indecomposable ordinals (named the...

sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets...

classified as open or closed: open classes (typically including nouns, verbs and adjectives) acquire new members constantly, while closed classes (such as...

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...

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...

n) = 1 implies gcd(ab, n) = 1, the set of classes coprime to n is closed under multiplication. Integer multiplication respects the congruence classes; that...

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

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are...

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

include the familiar arithmetic operations like addition, subtraction, multiplication, set operations like union, complement, intersection. Other examples are...