I'm not sure this statement is correct or not But I'm guess this statement is false.
statement)
Let the ring $R$ and its ideal $I$ and $J$ s.t. $I\subset J$
Then there is a subring(or ideal) $R_J(\simeq R/J)$ of the $R/I$ .
So Let me think about the polynomial quotient's ring $F[x]$ for a filed $F$
If we follow the same logical process,
For $f,g\in F[x]$ $s.t$ $f|g$(ie.$\langle g \rangle \subset \langle f \rangle$ )
There is a subring $F_f\simeq (F[x] / \langle f \rangle) $ whichi is a ideal of $F[x] / \langle g\rangle$
This would be also false statement.
But it looks like a definetly ture that
there is a ideal of $Q[x] / \langle (x-1)(x-2) \rangle$ who is a ismorphic with the $Q[x] / \langle x-1 \rangle$
This is contradict what I've known that the statement is false.
I'm really confused.
The question is
First)
Please clarify which one is correct among the statement, the case of the $F[x]$ and its example $Q[x]$.
Plus It would be thankful What I've missed.
Second) If the above statement is true,
Can we say that is correct when considering the commutative ring with unity?
(More generalized version)