Let $f: R_1 \rightarrow R_2$ be a surjective ring homomorphism.
Let $I_1$ be defined: $I_1=\{r_1\in R_1| f(r_1)\in I_2\}$, and I've showed that $I_1 \lhd R_1$. It's also known that $I_2 \lhd R_2$.
Prove: $R_1/ I_1 \cong R_2/I_2$
Can I just define an isomorphism?