Image of arbitrary ideal under quotient map

Question

Proposition 3.11 being the third isomorphism theorem for Rings.

I know that in a Ring there is a correspondence between ideals containing $I$ and the ideals of $R/I$. But I'm not sure about the image of an arbitrary ideal.

My guess is that $\varphi(J)=\varphi(J+I)$, still I feel uncomfortable with this statement since then $(R/I)/(J/I) \cong R/J$ which I know isn't right. So why is it that $\overline{J} = (I+J)/I$ and not $\overline{J} = J/I$?

Thanks in advanced for any help.

Answer 1

If $J\not\supseteq I$, writing $J/I$ doesn't make much sense. On the other hand $$ \{x+I:x\in J\}=\{x+I:x\in I+J\} $$ is true and $I+J\supseteq I$.

Your $\overline{J}$ is indeed $(I+J)/I$ and $$ \frac{R/I}{\overline{J}}=\frac{R/I}{(I+J)/I}\cong \frac{R}{I+J} $$