Boolean ring

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 (http://planetmath.org/CommutativeRing).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 (http://planetmath.org/BooleanLattice)).Given a Boolean ring $R$,define $x\wedge y=xy$ and $x\vee y=x+y+xy$ and $x^{\prime }=x+1$for all $x,y\in R$,then $(R,\wedge ,\vee {,}^{\prime },0,1)$ is a Boolean algebra.Given a Boolean algebra $(L,\wedge ,\vee {,}^{\prime },0,1)$,define $x\cdot y=x\wedge y$ and $x+y=(x^{\prime }\wedge y)\vee (x\wedge y^{\prime })$,then $(L,\cdot ,+)$ is a Boolean ring.In particular, the category of Boolean rings is isomorphic to the category of Boolean lattices.

Examples

As mentioned above, every Boolean algebra can be considered as a Boolean ring. In particular, if $X$ is any set, then the power set $𝒫(X)$ forms a Boolean ring, with intersection as multiplication and symmetric difference as addition.

Let $R$ be the ring ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ with the operations being coordinate-wise.Then we can check:

$(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)$

the four elements that form the ring are idempotent. So $R$ is Boolean.