Ring Zero is Unique

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Theorem

Let $\struct {R, +, \circ}$ be a ring.

Then the ring zero of $R$ is unique.


Proof 1

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

The result then follows from Identity of Group is Unique.

$\blacksquare$


Proof 2

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.

$\blacksquare$


Proof 3

Suppose $0$ and $0'$ are both ring zeroes of $\struct {R, +, \circ}$.

Then by Ring Product with Zero:

$0' \circ 0 = 0$ by dint of $0$ being a ring zero
$0' \circ 0 = 0'$ by dint of $0'$ being a ring zero.

So $0 = 0' \circ 0 = 0'$.

So $0 = 0'$ and there is only one ring zero of $\struct {R, +, \circ}$ after all.

$\blacksquare$