Boolean Ring

Definition¹

Let $R$ be a ring.

If $r \in R$ satisfies $r^2 = r$, then $r$ is called an idempotent element.
If every element of $R$ is idempotent, then $R$ is called a Boolean ring.

Explanation

A purist Korean rendering of 'Boolean ring' would be 'bulhwan', but it sounds so bad that the English pronunciation was used as is.

Projection in linear algebra has plenty of useful properties, so needless to say the same holds in generalized abstract algebra.

The most famous example of a Boolean ring is, of course, $$ (\left\{ \text{True}, \text{False} \right\} , \text{OR}, \text{AND} ) $$ which is also referred to as 'Boolean algebra'. As is well known, $$ \text{True AND True} = \text{True} \\ \text{False AND False} = \text{False} $$ so this ring is a Boolean ring. A more familiar example for students of mathematics is $\mathbb{Z}_{2}$, and naturally, a Boolean ring and $\mathbb{Z}_{2}$ are isomorphic.

Meanwhile, the following is known as a property of Boolean rings.

Theorem

A Boolean ring is a commutative ring.

Proof

For a Boolean ring $R$, if $a, b \in R$ then $(a+b) \in R$ and $$ (a + b)^2 = (a+b) $$ By the distributive law, $$ (a + b)^2 = (a+b)a + (a+b)b = a^2 + ba + ab + b^2 = (a+b) $$ Since $a^2 = a$ and $b^2 = b$, $$ a+ ba + ab + b = a+ b $$ Since $a$ and $b$ have additive inverses, rearranging gives $$ ba +ab = 0 $$ Adding $(-ba)$, the inverse of $ba$, to both sides yields $ab = -ba$, so $$ ab = (ab)^2 = (-ba)^2 = (ba)^2 = ba $$

■