Null Ring is Ring
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Theorem
Let $R$ be the null ring.
That is, let:
- $R := \struct {\set {0_R}, +, \circ}$
where ring addition and ring product are defined as:
\(\ds 0_R + 0_R\) | \(=\) | \(\ds 0_R\) | ||||||||||||
\(\ds 0_R \circ 0_R\) | \(=\) | \(\ds 0_R\) |
Then $R$ is a ring.
Proof
A null ring is a trivial ring.
So, by Trivial Ring is Commutative Ring, the result follows.
$\blacksquare$