The idempotent elements of Eisenstein Integers

Question

Let a+bω be an Eisenstein integer.
An idempotent element of $ \mathbb Z_n[\omega]$ is $(a+b\omega)^2 \equiv (a+b\omega)\pmod{n} $, where $\omega^2=-\omega-1$ But it follows that the idempotent element of this ring is always idempotent if and only if a is an element of n and b is equal to zero. Im having problems on proving this statement is true.

Answer 1

Expand $(a + wb)^2 - (a+b)$ to get the following:

$$(2ab - b)w + a2 - 2b - a \equiv 0 \pmod{n}$$

If $b \equiv 0$ and $a \equiv 0$ Then the above statement is true and so $(a+bw)$ is idempotent.