A problem about ring isomorphism?

Question

problem:let $R$ and $R'$ be commutative rings,and let $I⊆R$ and $I'⊆R'$ be ideals. if $f:R→R'$ is homomorphism with $f(I)⊆I'$,prove that : (i)$f_*:r+I→f(r)+I'$ is well-defined homomorphism $f_*:R/I→R'/I'$ (ii)If $f$ is isomorphism,then $f_*$ is also.

My attempt: For(i) I use condition $f(I)⊆I'$ to prove $f*$ is well-defined, and the homomorphism of $f_*$can derived by homomorphism $f$. For(ii) i don't know how to do it.Because i think (ii) is wrong. We know $ker f_*=\{r:f_*(r)=0 \}=\{r:f(r)+I'=I' \}=\{r:f(r)∈I' \}=f^{-1}(I')$.

If $ f_*$ is isomorphism,then $ker f_*=f^{-1}(I')=\{0\}$,but $f$ is also isomorphism,then $I'=0$.

Answer 1

Indeed, $$\ker f_* = \{r+I : r \in R,\, f(r)+I' = I'\} = \{r+I : r \in f^{-1}(I')\} = f^{-1}(I')/I$$ and note that $f_*$ is surjective if and only if for each $r' \in R'$ there exists $r \in R$ such that $r'+I' = f(r)+I'$, that is, if for each $r' \in R'$ there exists $r \in R$ such that $r'-f(r) \in I'$.

Thus, we see that

• $f_*$ is injective iff $I = f^{-1}(I')$; and
• $f_*$ is surjective iff $R' = \operatorname{im} f + I'$.