Boolean rings

Characteristic 2. $(x+x)^2=(2x)^2=4x^2=4x$, while $(x+x)^2=x+x=2x$ (since $x+x$ is also in $R$ and is idempotent). So $4x=2x$, hence $2x=0$.

Commutativity. $(x+y)^2=x+y$. Expanding (using the fact that addition is commutative in the ring structure, but multiplication may not be): $(x+y)^2=x^2+xy+yx+y^2=x+xy+yx+y$. Equating: $x+y=x+xy+yx+y$, so $xy+yx=0$, i.e. $yx=-xy=xy$ (since $-1=1$). Commutative.

Conclusion. Boolean rings are commutative and have characteristic $2$.