When do contractions respect ideal sums?

Question

Let $\varphi: R \rightarrow S$ be a homomorphism of commutative rings. Given two ideals $I, J \subseteq S$, when does the following equation hold: $$ \varphi^{-1}(I + J) = \varphi^{-1}(I) + \varphi^{-1}(J)$$

There are many examples which show that this isn't true in general, but are there any useful criteria when this holds after all?

Answer 1

If $\varphi$ is epic,then your equation holds. First,the rightside is include in the left.

• to see another side.for arbitrary $x \in \varphi^{-1}(I+J)$,$\varphi (x) \in I+J$.So there exists $a$ and $b$ such that $\varphi (x)=a+b$.so there exists $y$ and $z$ such that $\varphi (y)=a,\varphi(z)=b$ since $\varphi$ is epic.
• so we can get $\varphi (x)=\varphi(y)+\varphi(z)=\varphi(y+z)$. So $x-(y+z)$ lies in the kernel of $\varphi$,so there exists $c$ such that $x-(y+z)=c$,that is $x=(c+y)+z$,it is trivial to see that $c+y$ lies in $\varphi^{-1} I$.