Let $R$ be a Boolean ring with unity. Then show that $1 − 2a$ is a unit for all $a ∈ R$.

Question

Let $R$ be a Boolean ring with unity. Then show that $1 − 2a$ is a unit for all $a ∈ R$. I don't know how to do it. Any help is appereciated.

Answer 1

In a Boolean ring we have $a^2=a$ for every element $a$, so that:

$$2=1+1=\left(1+1\right)^{2}=1\cdot1+1\cdot1+1\cdot1+1\cdot1=\left(1+1\right)+\left(1+1\right)=2+2$$

This implies that $2=0$ and consequently $1-2a=1$ for every $a$.

Answer 2

In fact, for any ring $R$ at all, and any idempotent $a\in R$ (an element satisfying $a^2=a$, that is), you have $(1-2a)^2=1$, so $1-2a$ is a unit.

To go a little further afield, if $2$ is a unit in $R$, then for any element $b$ satisfying $b^2=1$, you have $\frac{1-b}{2}$ is an idempotent. So in rings where $2$ is invertible, you have a 1-1 correspondence between idempotents and "reflections."

It is bizarre to ask this question in a Boolean ring (since $2=0$ in such a ring, as mentioned by drhab), but it does guarantee that $a$ is idempotent.