The ideal which contains unit is whole ring.
$R$ : ring with unity, $J$ : ideal of $R$.
Suppose that unit $u \in J, \exists u\in R$.
Then, $x \in R \implies x \cdot u^{-1} \in R \implies (x \cdot u^{-1}) \cdot u \in J \implies x \in J \ (\forall x \in R)$.
Hence, $R \subseteq J$, and by definition of ideal, $J \subseteq R$.
$\therefore R = J$
Suppose that unit $u \in J, \exists u\in R$.
Then, $x \in R \implies x \cdot u^{-1} \in R \implies (x \cdot u^{-1}) \cdot u \in J \implies x \in J \ (\forall x \in R)$.
Hence, $R \subseteq J$, and by definition of ideal, $J \subseteq R$.
$\therefore R = J$
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