Ideal of Unit is Whole Ring

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

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

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

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

Let $$x \in R$$.

$$ $$ $$ $$

Thus $$R \subseteq J$$.

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