Null Ring is Ring with Unity
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Theorem
Let $R$ be the null ring.
Then $R$ is a ring with unity.
Proof
We have that $R$ is the null ring.
That is, by definition it has a single element, which can be denoted $0_R$, such that:
- $R := \struct {\set {0_R}, +, \circ}$
where ring addition and the ring product are defined as:
\(\ds 0_R + 0_R\) | \(=\) | \(\ds 0_R\) | ||||||||||||
\(\ds 0_R \circ 0_R\) | \(=\) | \(\ds 0_R\) |
Hence we have that the algebraic structure $\struct {\set {0_R}, \circ}$ is a trivial group.
Thus we see that:
- $\forall a, b \in R: a \circ b = 0_R$
Thus by definition, $0_R$ is a unity.
$\blacksquare$