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 $J = R$.

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

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

If $J$ contains the unity of $R$, then $J = R$.

Proof
Let $u \in J$, where $u \in U_R$.

Also by definition, we have $u^{-1} \in U_R$.

Let $x \in R$.

Thus $R \subseteq J$.

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

Proof of Corollary
Follows directly from the main result and the fact that Unity is a Unit.