Ring Zero is not Cancellable

Theorem
Let $\struct {R, +, \circ}$ be a ring which is not null.

Let $0$ be the ring zero of $R$.

Then $0$ is not a cancellable element for the ring product $\circ$.

Proof
$0$ is cancellable.

Let $a, b \in R$ such that $a \ne b$.

By definition of ring zero:
 * $0 \circ a = 0 = 0 \circ b$

By our supposition that $0$ is cancellable:
 * $a = b$

The result follows by Proof by Contradiction.