Boolean Ring

Definition ¹

Let’s call $R$ a ring.

If $r \in R$ satisfies $r^2 = r$, then $r$ is called an Idempotent Element.
If all elements of $R$ are idempotent, $R$ is called a Boolean Ring.

Explanation

Although ‘Boolean ring’ could be translated phonetically in Korean, the term sounds awkward, hence the English pronunciation was used directly.

The property of projection in linear algebra is known to be very useful, needless to say in generalized abstract algebra.

The most famous example of a Boolean ring is, of course, what is also referred to as ‘Boolean algebra’ $$ (\left\{ \text{True}, \text{False} \right\} , \text{OR}, \text{AND} ) $$ As well-known $$ \text{True AND True} = \text{True} \\ \text{False AND False} = \text{False} $$ thus, this ring becomes a Boolean ring. A more familiar example for mathematicians is $\mathbb{Z}_{2}$, which, of course, is isomorphic to the Boolean ring.

Meanwhile, the following property of the Boolean ring is known.

Theorem

The Boolean ring is a commutative ring.

Proof

For the Boolean ring $R$, if $a, b \in R$ then $(a+b) \in R$ and $$ (a + b)^2 = (a+b) $$ by the distributive law $$ (a + b)^2 = (a+b)a + (a+b)b = a^2 + ba + ab + b^2 = (a+b) $$ $a^2 = a$ and $b^2 = b$ hence $$ a+ ba + ab + b = a+ b $$ $a$ and $b$ exist as additive inverses, thus summarizing $$ ba +ab = 0 $$ Adding the inverse $(-ba)$ of $ba$ to both sides gives $ab = -ba$ hence $$ ab = (ab)^2 = (-ba)^2 = (ba)^2 = ba $$

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