# Boolean ring

An associative ring $K$ whose elements are all idempotent, i.e. $x^2=x$ for any $x\in K$. Any Boolean ring $K\neq0$ is commutative and is a subdirect sum of fields $\mathbf Z_2$ of two elements, and $x+x=0$ for all $x\in K$. A finite Boolean ring $K\neq0$ is a direct sum of fields $\mathbf Z_2$ and therefore has a unit element.

A Boolean ring is the ring version of a Boolean algebra, namely: Any Boolean algebra is a Boolean ring with a unit element under the operations of addition and multiplication defined by the rules

$$(x+y)=(x\cap Cy)\cup(y\cap Cx),\quad x\cdot y=x\cap y,$$

where $Cx$ is the complement of $x$. The zero and the unit of the ring are the same as, respectively, the zero and the unit of the algebra. Conversely, every Boolean ring with a unit element is a Boolean algebra under the operations $x\cup y=x+y+xy$, $x\cap y=x\cdot y$, $Cx=1+x$.

#### References

[1] | M.H. Stone, "The theory of representations for Boolean algebras" Trans. Amer. Math. Soc. , 40 (1936) pp. 37–111 |

[2] | I.I. Zhegalkin, "On the technique of computation of propositions in symbolic logic" Mat. Sb. , 34 : 1 (1927) pp. 9–28 (In Russian) (French abstract) |

[3] | D.A. Vladimirov, "Boolesche Algebren" , Akademie Verlag (1978) (Translated from Russian) |

[4] | R. Sikorski, "Boolean algebras" , Springer (1969) |

#### Comments

The operation $x+y=(x\cap Cy)\cup(y\cap Cx)$ is known as the symmetric difference. Think of the Boolean algebra of all subsets of a given set under union, intersection and complement to interpret these formulas.

#### References

[a1] | S. Rudeanu, "Boolean functions and equations" , North-Holland (1974) |

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Boolean ring.

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