The only boolean ring that is an integral domain is Z/2Z

A Boolean ring that is an integral domain is Z/2Z

We will prove that the only boolean ring that is an integral domain is

\mathbb{Z}/2\mathbb{Z}

Let

R

be a boolean ring which is an integral domain. Boolean rings are commutative, as we have seen here. Take the elements

a, 1-a \in R

which are not equal to

0

and

1

. Then we get:

\begin{align*} a(1-a) = a - a^2 = 0, \end{align*}

since

a^2 = a

. We took for the assumption that our boolean ring is an integral domain, so this implies that

a

and

1-a

are no zero divisors. This leads us that

a = 0

1 - a = 0

, which implies that

R = \{0,1\}

and this is exactly

\mathbb{Z}/2\mathbb{Z}

The only boolean ring that is an integral domain is Z/2Z.