examples of ring of sets

Every field of sets is a ring of sets. Below are some examples of rings of sets that are not fields of sets.

1. Let $A$ be a non-empty set containing an element $a$. Let $ℛ$ be the family of subsets of $A$ containing $a$. Then $ℛ$ is a ring of sets, but not a field of sets, since $\{a\}\in ℛ$, but $A-\{a\}\notin ℛ$.

2. The collection of all open sets of a topological space is a ring of sets, which is in general not a field of sets, unless every open set is also closed. Likewise, the collection of all closed sets of a topological space is also a ring of sets.

3. A simple example of a ring of sets is the subset $\{\{a\},\{a,b\}\}$of $2^{\{a,b\}}$. That this is a ring of sets follows from theobservations that $\{a\}\cap \{a,b\}=\{a\}$ and $\{a\}\cup \{a,b\}=\{a,b\}$. Note that it is not a field of sets because thecomplement of $\{a\}$, which is $\{b\}$, does not belong to the ring.

4. Another example involves an infinite set. Let $A$ be an infinite set. Let $ℛ$ be the collection of finite subsets of $A$. Since the union and the intersection of two finite set are finite sets, $ℛ$ is a ring of sets. However, it is not a field of sets, because the complement of a finite subset of $A$ is infinite, and thus not a member of $ℛ$.