Commutativity of Boolean Rings (7.1.15-16)

Dummit and Foote Abstract Algebra, section 7.1, exercises 15-16:

15. A ring $R$ is a Boolean ring if $a^2=a$ for all $a∈R$. Prove that every Boolean ring is commutative.
16. Prove that the only Boolean integral domain is $\mathbb{Z}/2\mathbb{Z}$.

Proof: (15) Take any $a,b∈R$. Note that we necessarily have $(-x)^2=-x$ for any $x∈R$ as well as $(-x)^2=(-x)(-x)=x^2=x$, so that $-x=x$. Now, notice that $(a+b)^2=a+b$ by the Boolean property. By distributing, we have $(a+b)^2=a^2+ab+ba+b^2=a+ab+ba+b$. Comparing the two, we have $ab+ba=0$, so that $ab=-ba=ba$.$~\square$

(16) For any $a$ in a Boolean integral domain, we have $a(a-1)=a^2-a=a-a=0$, so that either $a=0$ or $a-1=0$ implying $a=1$.$~\square$