Null Ring is Ideal

Theorem
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Then the null ring $\struct {\set {0_R}, +, \circ}$ is an ideal of $R$.

Proof
From Null Ring and Ring Itself Subrings, $\set {0_R}$ is a subring of $\struct {R, +, \circ}$.

Also:
 * $\forall x \in \struct {R, +, \circ}: x \circ 0_R = 0_R = 0_R \circ x \in \set {0_R}$

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