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

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 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.

4.
Another example involves an infinite set^{}. Let $A$ be an infinite set. Let $\mathcal{R}$ be the collection of finite subsets of $A$. Since the union and the intersection^{} of two finite set^{} are finite sets, $\mathcal{R}$ 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 $\mathcal{R}$.
Title  examples of ring of sets 

Canonical name  ExamplesOfRingOfSets 
Date of creation  20130322 15:47:52 
Last modified on  20130322 15:47:52 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  8 
Author  rspuzio (6075) 
Entry type  Example 
Classification  msc 03E20 
Classification  msc 28A05 