Ideal of Unit is Whole Ring

Theorem
Let $$\left({R, +, \circ}\right)$$ be a ring with unity.

Let $$S$$ be an ideal of $$R$$.

If $$S$$ contains a unit of $$R$$, then $$S = R$$.

Proof
Let $$u \in S$$, where $$u \in U_R$$ (and also by definition where $$u^{-1} \in U_R$$).

Let $$x \in R$$.

Thus $$R \subseteq S$$.

As $$S \subseteq R$$ by definition, it follows that $$S = R$$.