Ring Zero is Unique/Proof 3
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Theorem
Let $\struct {R, +, \circ}$ be a ring.
Then the ring zero of $R$ is unique.
Proof
Suppose $0$ and $0'$ are both ring zeroes of $\struct {R, +, \circ}$.
Then by Ring Product with Zero:
- $0' \circ 0 = 0$ by dint of $0$ being a ring zero
- $0' \circ 0 = 0'$ by dint of $0'$ being a ring zero.
So $0 = 0' \circ 0 = 0'$.
So $0 = 0'$ and there is only one ring zero of $\struct {R, +, \circ}$ after all.
$\blacksquare$
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties