Null Ring is Ideal

Theorem
Let $\left({R, +, \circ}\right)$ be a ring whose zero is $0_R$.

Then the null ring $\left({\left\{{0_R}\right\}, +, \circ}\right)$ is an ideal of $R$.

Proof
From Null Ring and Ring Itself Subrings, $\left\{{0_R}\right\}$ is a subring of $\left({R, +, \circ}\right)$.

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

thus fulfilling the condition for $\left\{{0_R}\right\}$ to be an ideal of $R$.