Null Ring and Ring Itself Subrings

Theorem
In any ring $$R$$, the null ring and $$R$$ itself are subrings of $$R$$.

Proof

 * $$R$$ is a ring and $$R \subseteq R$$, so it follows trivially that $$R$$ is a subring of $$R$$.


 * The null ring $$\left({\left\{{0_R}\right\}, +, \circ}\right)$$ is always a ring, and a subset of $$R$$, so has to be a subring of $$R$$.