Corollary. About Boolean ring [math-HS0H]

Let $A$ be a Boolean ring (see nLab) and $X = \operatorname{Spec} A$.

1. Every prime ideal of $A$ is a maximal ideal. Details
2. $\kappa(x) := \text{Frac}(A/x) = \mathbb{F}_2$ for all $x \in X$. Details
3. $A$ has characteristic 2. Because for all $x \in A$ we have $$x + x = (x + x) * (x + x) = x*x + x*x + x*x + x*x = x+x+x+x$$ deduces that $2x = x + x = 0$. Notice that, this also say $x = -x$ for all $x \in A$.
4. For each $\varphi : A \to \mathbb{F}_2$ we can define $$\varphi \mapsto \operatorname{ker} \varphi$$ these maps form a bijection between hom-set $\text{Hom}(A, \mathbb{F}_2)$ and $\operatorname{Spec} A$. Details