Ring Zero is not Cancellable

Theorem

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

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

By definition of ring zero:

$0 \circ a = 0 = 0 \circ b$

But if $0$ were cancellable, then $a = b$.

The result follows by Proof by Contradiction.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 21$