A Boolean ring is a ring $R$ that has a multiplicative identity, and in which every element is idempotent, that is,

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 $x^{{\prime}}=x+1$ for all $x,y\in R$, then $(R,\land,\lor,\phantom{i}^{{\prime}},0,1)$ is a Boolean algebra. Given a Boolean algebra $(L,\land,\lor,\phantom{i}^{{\prime}},0,1)$, define $x\cdot y=x\land y$ and $x+y=(x^{{\prime}}\land y)\lor(x\land y^{{\prime}})$, then $(L,\cdot,+)$ is a Boolean ring. In particular, the category of Boolean rings is isomorphic to the category of Boolean lattices.