Boolean Ring - Properties of Boolean Rings

Properties of Boolean Rings

Every Boolean ring R satisfies x ⊕ x = 0 for all x in R, because we know

x ⊕ x = (x ⊕ x)2 = x2 ⊕ 2x2 ⊕ x2 = x ⊕ 2x ⊕ x = x ⊕ x ⊕ x ⊕ x

and since <R,⊕> is an abelian group, we can subtract x ⊕ x from both sides of this equation, which gives x ⊕ x = 0. A similar proof shows that every Boolean ring is commutative:

x ⊕ y = (x ⊕ y)2 = x2 ⊕ xy ⊕ yx ⊕ y2 = x ⊕ xy ⊕ yx ⊕ y

and this yields xy ⊕ yx = 0, which means xy = yx (using the first property above).

The property x ⊕ x = 0 shows that any Boolean ring is an associative algebra over the field F₂ with two elements, in just one way. In particular, any finite Boolean ring has as cardinality a power of two. Not every associative algebra with one over F₂ is a Boolean ring: consider for instance the polynomial ring F₂.

The quotient ring R/I of any Boolean ring R modulo any ideal I is again a Boolean ring. Likewise, any subring of a Boolean ring is a Boolean ring.

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R/P is an integral domain and also a Boolean ring, so it is isomorphic to the field F₂, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Boolean rings are von Neumann regular rings.

Boolean rings are absolutely flat: this means that every module over them is flat.

Every finitely generated ideal of a Boolean ring is principal (indeed, (x,y)=(x+y+xy)).