Show 0 and 1 are the only elements satisfying the identity $a^2=a$ for an integral domain. Give a ring that has three elements that satisfy $a^2=a$.

Question

Let $R$ be an integral domain. Show that $0$ and $1$ are the only elements satisfying the identity $a^2=a$. Give an example of a ring and at least three elements in that ring that satisfy $a^2 = a$.

For the first part, how would I go about proving this?

For the second part, aren't the elements $[0]_6,[1]_6,[3]_6\in \Bbb Z/6\Bbb Z$ an example of elements from the ring $\Bbb Z/6\Bbb Z$ that satisfy $a^2 = a$ because $[0^2]_6=[0]_6$ ,$[1^2]_6=[1]_6$, and $[3^2]_6=[9]_6=[3]_6$?

Answer 1

Hint: $a^2 = a \implies a(a-1)=0$