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# 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. 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.

# 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.

## Mathematics Subject Classification

06E99*no label found*03G05

*no label found*

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