Ring is Ideal of Itself

Theorem
Let $\struct {R, +, \circ}$ be a ring.

Then $R$ is an ideal of $R$.

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

Also:
 * $\forall x, y \in \struct {R, +, \circ}: x \circ y \in R$

thus fulfilling the condition for $R$ to be an ideal of $R$.