Ring generated by

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Here ring refers to a ring of sets

Definition

Given any class of sets A, there exists a unique ring R0 such that ER0 and such that if R is any ring with ER and RR0 then R0R

We call R0 the ring generated by A, often denoted R(A)

Proof

Since P(X) (where A is a collection of subsets of X) is a ring (infact an algebra) we know that a ring containing A exists.

Since the intersection of any collection of rings is a ring (see the theorem here), it is clear that the intersection of all rings containing A is the required ring R0.

Important theorems

Every set in R(A) can be finitely covered by sets in A

[Expand]

If A is any class of sets, then every set in R(A) can be covered by a finite union of sets in A