# 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^{} $\mathcal{P}(X)$ 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.

Title | Boolean ring |
---|---|

Canonical name | BooleanRing |

Date of creation | 2013-03-22 12:27:28 |

Last modified on | 2013-03-22 12:27:28 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 24 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 06E99 |

Classification | msc 03G05 |

Related topic | Idempotency |

Related topic | BooleanLattice |

Related topic | BooleanIdeal |