Ring Zero is not Cancellable
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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): $\S 21$