Ring Zero is Idempotent

Theorem
Let $\left({R, +, \circ}\right)$ be a ring with ring zero $0_R$.

Then $0_R$ is an idempotent element of $R$, that is:

$0_R\circ0_R = 0_R$.

Proof
By Ring Product with Zero (applied to $0_R$):

$0_R\circ0_R = 0_R$,

which was to be proven.