Null Ring is Trivial Ring

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Let $R$ be the null ring.

Then $R$ is a trivial ring.


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:

\(\ds 0_R + 0_R\) \(=\) \(\ds 0_R\)
\(\ds 0_R \circ 0_R\) \(=\) \(\ds 0_R\)

Consider the operation $+$.

By definition, the algebraic structure $\struct {\set {0_R}, +}$ is a trivial group.


$\forall a, b \in R: a \circ b = 0_R$

Thus by definition, $R$ is a trivial ring.