# Two ideals and their quotient rings (inclusion relationship)

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.

• If I understood your notations correct, $R_I=\{ 1,3\}$ is not a subring of $Z /\langle 4 \rangle$, becuase $1+1=2\not\in R_I$. – Alex Ravsky Jul 27 at 4:16