What Are Examples of Infinite Boolean Rings?

A Boolean ring is a ring in which every element is idempotent: multiplying an element by itself leaves it unchanged. There are many infinite examples, and a useful way to construct them is to use sets. The main example works whether or not your definition of ring requires a multiplicative identity.

The defining property

A ring $R$ is Boolean if

$$x^2=x \quad\text{for every }x\in R.$$

Under the usual ring conventions, every Boolean ring is commutative, and every element is its own additive inverse (so $2x=0$ for all $x$). Thus every nonzero Boolean ring has characteristic 2. In particular, addition behaves like exclusive OR: adding an element to itself gives zero. Some authors require a ring to have a multiplicative identity, while others allow rings without one. The examples below identify which convention they meet.

The power set of an infinite set

Let $S$ be any infinite set, and take all subsets of $S$ as the elements of a ring. Define addition and multiplication by

$$A+B=A\mathbin{\triangle}B,\qquad AB=A\cap B,$$

where $A\mathbin{\triangle}B$ is the symmetric difference: the elements belonging to exactly one of $A$ and $B$.

These operations satisfy the ring laws. Symmetric difference is associative and commutative, has the empty set as zero, and each set is its own additive inverse. Intersection is associative and distributes over symmetric difference. For every subset $A$,

$$A^2=A\cap A=A,$$

so the ring is Boolean. It is infinite because an infinite set has infinitely many distinct subsets. Its multiplicative identity is $S$, since $A\cap S=A$.

Worked example

Take $S=\{1,2,3,4,\ldots\}$, $A=\{1,2\}$, and $B=\{2,4\}$. Then

$$A+B=A\mathbin{\triangle}B=\{1,4\},\qquad AB=A\cap B=\{2\}.$$

The repeated product of $A$ is $A^2=A\cap A=\{1,2\}=A$. Also, $A+A=\varnothing$, because every element of $A$ occurs in both copies and therefore cancels under symmetric difference.

Other infinite examples

Finite-cofinite subsets. On an infinite set $S$, take the subsets that are either finite or have finite complement. With symmetric difference and intersection as operations, these subsets form a Boolean ring. The operations preserve the finite-or-cofinite property, and the same calculation $A\cap A=A$ proves the Boolean condition. This ring is infinite and has identity $S$.

Finite subsets only. The collection of all finite subsets of an infinite set $S$, with the same operations, is also an infinite Boolean ring. It is closed under symmetric difference and intersection. It does not have a multiplicative identity when $S$ is infinite: an identity would have to contain every finite subset, and thus every element of $S$, so it would have to be $S$, which is not finite. This example is useful only if your convention permits rings without an identity.

Functions into the two-element field. Let $I$ be infinite and consider all functions $f:I\to\mathbb{F}_2$, with addition and multiplication performed pointwise. Since each value in $\mathbb{F}_2=\{0,1\}$ satisfies $a^2=a$, every function satisfies $f^2=f$. This ring is unital, with the constant-one function as identity. It is another description of the power-set example: associate to each function the subset of indices where its value is 1.

Why these examples are Boolean

In the set construction, multiplication is intersection, and intersection is idempotent: $A\cap A=A$. In the function construction, multiplication is pointwise, and both possible values, 0 and 1, are idempotent. The defining property therefore follows directly from the operations rather than from a more complicated calculation.

The set examples also illustrate the relationship between Boolean rings and Boolean algebras. The ring operations encode familiar set operations: addition corresponds to symmetric difference, and multiplication corresponds to intersection. For a unital Boolean ring, the multiplicative identity corresponds to the whole set; complements can be expressed using the ring operations and the identity.

Common mistakes

• Using ordinary union as addition. Union is not the ring addition for the power-set example. Symmetric difference is needed so that $A+A=\varnothing$.
• Forgetting to specify the operations. A collection of subsets is not automatically a ring; state which operations are used.
• Assuming every example has an identity. The power set and finite-cofinite examples are unital, but the finite-subsets example is not when the underlying set is infinite.
• Confusing finite with finite-cofinite. The finite subsets do not include the whole infinite set. Adding cofinite subsets gives the finite-cofinite ring, which does contain the identity $S$.