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.

Let be a nonempty collection of sets. Then is a 𝜎-ring if:

1. Closed under countable unions: if for all
2. Closed under relative complementation: if

These two properties imply: whenever are elements of

This is because

Every 𝜎-ring is a δ-ring but there exist δ-rings that are not 𝜎-rings.

If the first property is weakened to closure under finite union (that is, whenever ) but not countable union, then is a ring but not a 𝜎-ring.

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.

A 𝜎-ring that is a collection of subsets of induces a 𝜎-field for Define Then is a 𝜎-field over the set - to check closure under countable union, recall a -ring is closed under countable intersections. In fact is the minimal 𝜎-field containing since it must be contained in every 𝜎-field containing

• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• Join (sigma algebra) – Algebraic structure of set algebraPages displaying short descriptions of redirect targets
• 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
• Measurable function – Kind of mathematical function
• Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• Sample space – Set of all possible outcomes or results of a statistical trial or experiment
• 𝜎 additivity – Mapping function
• σ-algebra – Algebraic structure of set algebra
• 𝜎-ideal – Family closed under subsets and countable unions

• Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses 𝜎-rings in development of Lebesgue theory.

Families of sets over
Is necessarily true of or, is closed under:	Directed by									F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if	only if
𝜆-system (Dynkin System)				only if			only if or they are disjoint			Never
Ring (Order theory)
Ring (Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra (Field)										Never
𝜎-Algebra (𝜎-Field)										Never
Dual ideal
Filter				Never	Never
Prefilter (Filter base)				Never	Never
Filter subbase				Never	Never
Open Topology							(even arbitrary )			Never
Closed Topology						(even arbitrary )				Never
Is necessarily true of or, is closed under:	directed downward	finite intersections	finite unions	relative complements	complements in	countable intersections	countable unions	contains	contains	Finite Intersection Property
Additionally, a semiring is a π-system where every complement is equal to a finite disjoint union of sets in A semialgebra is a semiring where every complement is equal to a finite disjoint union of sets in are arbitrary elements of and it is assumed that