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

1. 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.
2. 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 the observations that $\{a\} \cap \{a,b\} = \{a\}$ and $\{a\} \cup \{a,b\} = \{a,b\}$ Note that it is not a field of sets because the complement of $\{a\}$ which is $\{b\}$ does not belong to the ring.
3. 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 $R$