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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 ...
In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} and { 3 , 4 } {\displaystyle \{3,4\}} is { 1 , 2 , 4 ...
The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a + b , then c n = a n + b n for every n in N .
The set D of finite elements of k is a valuation ring. The set of elements x such that x ∈ D and x −1 ∉ D is the set of infinitesimal elements; and an element x such that x ∉ D and x −1 ∈ D is called infinite. The ring F of finite elements of a hyperreal field *R (an ordered field containing the real numbers) is a valuation ring of *R.
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.
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𝜎-rings can be used instead of 𝜎-fields (𝜎-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every 𝜎-field is also a 𝜎-ring, but a 𝜎-ring need not be a 𝜎-field.