Ring Zero is not Cancellable
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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