Ring Zero is Unique

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

Then the ring zero of $R$ is unique.

Proof
The ring zero is, by definition of a ring, the identity element of the additive group $\left({R, +}\right)$.

The result the follows from Identity is Unique.

Alternatively, from Ring Product with Zero we have that the ring zero of $R$ is indeed a zero element, as suggested by its name.

The result then follows from Zero Element is Unique.