Null Ring is Trivial Ring

Theorem
Let $R$ be the null ring.

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:
 * $R := \struct {\set {0_R}, +, \circ}$

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 $+$.

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

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

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

The fact that $R$ is a commutative ring follows from Trivial Ring is Commutative Ring.