Null Ring is Ring with Unity

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:

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.