Alternative proof of '$I$ is maximal iff $R/I$ is a field'

Question

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.

Answer 1

Suppose that $F$ is a field, and $I$ is a non-zero ideal in $F$; show that $1_F\in I$ and then that $I=F$. The second equivalence is from the fourth isomorphism theorem for rings.

Answer 2

by using answer @Brian M.scott

1. Suppose that $F$ is a field,and $I$ is a non-zero ideal in $F$, if $x \in I$ then since $x^{-1}$ is in $F$ and since $I$ is ideal then,$$xx^{-1} \in I \rightarrow 1_{F} \in I$$ so, $I=R$

2. if $J$ be ideal containig $I$ and $J \neq R$ then $\frac{J}{I}$ is an ideal of $\frac{R}{I}$, since $\frac{R}{I}$ is field then by 1 $\frac{J}{I}$ is $\frac{R}{I}$ or ${0}=I$ then $J =I$ or $J=R$