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

Aiming for a contradiction, suppose $0$ 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.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains