False statement
Let the ring $R$ and its ideals $I$ and $J$ s.t. $I\subset J$.
$R/J$ is a ideal (or subring) of the $R/I$ ?
Since $R/J$ is not a subset of the $R/I$ (actually $R/J$ is a isomorphic with some quotient ring of the $R/I$, the above statement is surely incorrect.
For example $Q[x]/\langle(x-1) \rangle $ is not a subset of the $Q[x]/\langle(x-1)(x-2) \rangle$
(When we taking the $I (= \langle (x-1)(x-2) \rangle) \subset J(= \langle (x-1) \rangle) $)
The other example $Z/\langle 2 \rangle $ is not a subset of the $Z /\langle 4 \rangle$ (When we taking the $I (= \langle 4 \rangle) \subset J(= \langle 2 \rangle) $)
Question
Let the ring $R$ and its ideal $I$ and $J$ s.t. $I\subset J$
Let the $R_I$ be a subring or ideal of $R/I$
Then Does $R_I$ exist $s.t.$ isomorphic with the $R/J$ ?
(I.E. I want to know existence of $R/I$'s subring or ideal who is ismorphic with the $R/J$ )
It looks like a true when we considering the above two examples.(It's just my thoght.)
$Q \times \{0\}(\simeq Q[x]/\langle(x-1) \rangle \simeq Q )$ is a subring or ideal of the $Q[x]/\langle(x-1)(x-2) \rangle$ .
$\{ 1,3\}(\simeq Z/\langle 2 \rangle \simeq Z_2) $ is a subring or ideal of the $Z /\langle 4 \rangle$.
But I'm not sure this is right or not.
(Cause I couldn't find the other counterexmaples and don't know how to prove. :()
What do you think about that?
Any help would be appreciated.
Thanks for reading my post.