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If X is any set, then the power set of X (the family of all subsets of X) forms a ring of sets in either sense.. If (X, ≤) is a partially ordered set, then its upper sets (the subsets of X with the additional property that if x belongs to an upper set U and x ≤ y, then y must also belong to U) are closed under both intersections and unions.
A ring is a set R equipped with two binary operations [a] + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms: [1] [2] [3] R is an abelian group under addition, meaning that: (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative). a + b = b + a for all a, b in R (that ...
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of . By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a 𝜎-algebra .
Examples of multiplicative sets include: the set-theoretic complement of a prime ideal in a commutative ring; the set {1, x, x 2, x 3, ...}, 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.
In mathematics, a nonempty collection of sets is called a 𝜎-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
For a semi-ring , the set of all finite unions of sets in is the ring generated by : = {: = =,} (One can show that () is equal to the set of all finite disjoint unions of sets in ). A content μ {\displaystyle \mu } defined on a semi-ring S {\displaystyle S} can be extended on the ring generated by S . {\displaystyle S.}