Ring Zero is Idempotent

Theorem

Let $\struct {R, +, \circ}$ be a ring whose ring zero is $0_R$.

Then $0_R$ is an idempotent element of $R$ under the ring product $\circ$:

$0_R \circ 0_R = 0_R$

Proof

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

$0_R \circ 0_R = 0_R$

which was to be proven.

$\blacksquare$