Is there any maximal ideal that contains the unity element of a ring?

Question

I know that if an ideal $U=\langle1\rangle$, then $U$ becomes equal to $R$, the ring. So my thinking converges to the fact that there is no $ \mathbf{ proper}$ maximal ideal containing $1$. But do I prove it?

Answer 1

This is trivial to prove via contraposition. If an ideal contains $1$, it is the whole ring and is thus not maximal. Therefore, if an ideal is maximal, it does not contain $1$.

Answer 2

Once an ideal contains the rings unity, in fact once it contains a unit, it is the whole ring. Because ideals are closed under (left) multiplication.