# 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 $x\land y=xy$ 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.

## 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 ${\cal P}(X)$ 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:

 $\displaystyle(1,1)\times(1,1)$ $\displaystyle=$ $\displaystyle(1,1)$ $\displaystyle(1,0)\times(1,0)$ $\displaystyle=$ $\displaystyle(1,0)$ $\displaystyle(0,1)\times(0,1)$ $\displaystyle=$ $\displaystyle(0,1)$ $\displaystyle(0,0)\times(0,0)$ $\displaystyle=$ $\displaystyle(0,0)$

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

Title Boolean ring BooleanRing 2013-03-22 12:27:28 2013-03-22 12:27:28 yark (2760) yark (2760) 24 yark (2760) Definition msc 06E99 msc 03G05 Idempotency BooleanLattice BooleanIdeal