**Boolean Rings**

Every Boolean algebra (A, ∧, ∨) gives rise to a ring (*A*, +, ·) by defining *a* + *b* := (*a* ∧ ¬*b*) ∨ (*b* ∧ ¬*a*) = (*a* ∨ *b*) ∧ ¬(*a* ∧ *b*) (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and *a* · *b* := *a* ∧ *b*. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that *a* · *a* = *a* for all *a* in *A*; rings with this property are called Boolean rings.

Conversely, if a Boolean ring *A* is given, we can turn it into a Boolean algebra by defining *x* ∨ *y* := *x* + *y* + (*x* · *y*) and *x* ∧ *y* := *x* · *y*. Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map *f* : *A* → *B* is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The categories of Boolean rings and Boolean algebras are equivalent.

Read more about this topic: Boolean Algebra (structure)

### Famous quotes containing the word rings:

“We will have *rings* and things, and fine array,

And kiss me, Kate, we will be married o’ Sunday.”

—William Shakespeare (1564–1616)