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A Boolean ring is a ring $R$ that has a multiplicative identity, and in which every element is idempotent, that is, $$x^2=x\text{ for all }x\in R.$$ Boolean rings are necessarily commutative. Also, if $R$ is a Boolean ring, then $x=-x$ for each $x\in R$
Boolean rings are equivalent to Boolean algebras (or Boolean lattices). Given a Boolean ring $R$ define $x \land y = xy$ and $x \lor y = x + y + xy$ and $x'=x+1$ for all $x,y\in R$ then $(R,\land,\lor,\phantom{i}',0,1)$ is a Boolean algebra. Given a Boolean algebra $(L,\land,\lor,\phantom{i}',0,1)$ define $x\cdot y = x \land y$ and $x + y = (x' \land y) \lor (x \land y')$ then $(L,\cdot,+)$ is a Boolean ring. In particular, the category
of Boolean rings is isomorphic to the category of Boolean lattices.
As mentioned above, every Boolean algebra can be considered as a Boolean ring. In particular, if $X$ is any set, then the power set ${\cal P}(X)$ forms a Boolean ring, with intersection as multiplication and symmetric difference as addition.
Let $R$ be the ring $\Z_2\times\Z_2$ with the operations being coordinate-wise. Then we can check: \begin{eqnarray*} (1,1)\times(1,1)&=&(1,1)\\ (1,0)\times(1,0)&=&(1,0)\\ (0,1)\times(0,1)&=&(0,1)\\ (0,0)\times(0,0)&=&(0,0) \end{eqnarray*}the four elements that form the ring are idempotent. So $R$ is Boolean.
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