Idempotent Elements of a Commutative Ring

Question

I have to prove this statement and I'm a bit unsure how to go about it:

Show that the set of all idempotent elements of a commutative ring is closed under multiplication. Furthermore, find all the idempotent elements in the ring ${\bf Z}_6 \times {\bf Z}_{12}$.

So, I know that the proof should start off with saying that an element a of a ring $R$ is idempotent if $a^2 = a$. I'm not sure how to show that the set of all idempotent elements is closed under multiplication. Any help with the proof would be appreciated.

Answer 1

Hint:

(1)It's a commutative ring.

(2)$(a,b)(a,b)=(a^2,b^2)$

Answer 2

Let two elements of the ring be idempotent and see what happens when you square their product.