Order, lattices, ordered algebraic structures Boolean algebras Theorem

Equivalence of Boolean rings and Boolean algebras

Builds on Boolean algebra Boolean ring Symmetric difference

Statement

Let $(B, \wedge, \vee, \neg, 0, 1)$ be a Boolean algebra. Put $a + b = (a \wedge \neg b) \vee (\neg a \wedge b)$ and $a \cdot b = a \wedge b$ . Then $(B, +, \cdot, 0, 1)$ is a Boolean ring.
Let $(R, +, \cdot, 0, 1)$ be a Boolean ring. Put $a \wedge b = ab$ , $a \vee b = a + b + ab$ and $\neg a = 1 + a$ . Then $(R, \wedge, \vee, \neg, 0, 1)$ is a Boolean algebra.
The two constructions are inverse to each other.
A map between Boolean algebras is a homomorphism if and only if it is a unital ring homomorphism of the matching rings.

Remarks

The proof is a direct check of axioms. In a Boolean ring, $(a + b)^2 = a + b$ gives $ab + ba = 0$ , and $a = -a$ follows from $(a + a)^2 = a + a$ , so the ring is commutative of characteristic $2$ ; this makes the lattice laws routine. In the other direction, the key point is that $+$ is associative, which reduces to the distributive laws and De Morgan's laws. For the power set of a set $X$ , the ring addition is the symmetric difference $A \mathbin{\triangle} B$ and the product is $A \cap B$ . In categorical terms, the theorem gives an isomorphism between the category of Boolean algebras and the category of Boolean rings.

Going deeper

For a student

The ring view brings tools of commutative algebra to Boolean algebras. Ideals of the ring are exactly the ideals of the Boolean algebra, and prime ideals are maximal, with quotient the field $\F_2$ . So ultrafilters, which are complements of maximal ideals, correspond to ring homomorphisms onto $\F_2$ , and the Stone space of the algebra is the prime spectrum of the ring. This is the bridge used in one standard proof of Stone's representation theorem.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Boolean algebra concept
Boolean ring concept
Symmetric difference concept
Equivalence of Boolean rings and Boolean algebras statement

Tag	`PQEPV7F2RV`
URI	`m2://PQEPV7F2RV`
Kind	statement, theorem
Writing	W0 stub
MSC	06E20, 06E05
Status	This copy of the site has no build data. Run `m2 status PQEPV7F2RV`.
About	Boolean algebra, Boolean ring
Uses	Boolean algebra, Boolean ring, Symmetric difference
Changed	2 November 2026