Null Ring is Trivial Ring

Theorem
Let $R$ be the null ring.

That is, let:
 * $R := \left({\left\{{0_R}\right\}, +, \circ}\right)$

where ring addition and the ring product are defined as:


 * $0_R + 0_R = 0_R$
 * $0_R \circ 0_R = 0_R$

Then $R$ is a trivial ring and therefore a commutative ring.

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:
 * $R := \left({\left\{{0_R}\right\}, +, \circ}\right)$

where ring addition and the ring product are defined as:


 * $0_R + 0_R = 0_R$
 * $0_R \circ 0_R = 0_R$

Consider the operation $+$:

$\left({\left\{{0_R}\right\}, +}\right)$ is a group.

This follows from the definition of the trivial group.

We have that:
 * $\forall a \in R: a \circ a = 0_R$

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

The fact that $R$ is a commutative ring a consequence of Trivial Ring is Commutative Ring.