Ring Zero is Idempotent

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Theorem

Let $\left({R, +, \circ}\right)$ 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.

$\blacksquare$