Boolean ring
One example of a Boolean ring is the power set of any set "X", where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can also consider the set of all finite or cofinite subsets of "X", again with symmetric difference and intersection as operations. More generally with these operations any field of sets is a Boolean ring. By Stone's representation theorem every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).