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

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