For any commutative ring $R$ and an ideal $I$ of $R$, $I \neq R$, show that $I$ is a maximal ideal iff $R/I$ is a field.
I write my own proof and it checks with the 'traditional' proof which makes use of an element that is not in $I$. When I check the notes, my teacher gives this proof:
$R/I$ is a field
$\Leftrightarrow$ the only ideals in $R/I$ are zero ideal and itself
$\Leftrightarrow$ the only ideals in $R$ containing $I$ are $I$ and $R$
$\Leftrightarrow$ $I$ is maximal
The first and second equivalence relations are not so obvious for me. Can anyone provide some hints? Thanks in advance.