A question about surjective module-homomorphisms

[steenis]

I have the following question about surjective module-homomorphisms.

Let $f:A \longrightarrow B$ be a surjective $R$-homomorphism between $R$-modules $A$ and $B$.
Let $S, T$ be submodules of $A$ and let $X, Y$ be submodules of $B$.

I can prove that in general
$$f(S+T)=f(S)+f(T)$$
and in general
$$f^{-1}(X)+ f^{-1}(Y) \subseteq f^{-1}(X+Y)$$
If $f$ is surjective we can combine these two, and we have
$$f(f^{-1}(X)+ f^{-1}(Y))=f f^{-1}(X)+f f^{-1}(Y)=X+Y$$
But I need this:
$$f^{-1}(X)+ f^{-1}(Y)=f^{-1}(X+Y)$$
given that $f$ is surjective.

I cannot find the proof and I do not know if it is true, can somebody help me to prove this or give a counterexample or give a reference?

 

[Opalg]

I have the following question about surjective module-homomorphisms.

Let $f:A \longrightarrow B$ be a surjective $R$-homomorphism between $R$-modules $A$ and $B$.
Let $S, T$ be submodules of $A$ and let $X, Y$ be submodules of $B$.

I can prove that in general
$$f(S+T)=f(S)+f(T)$$
and in general
$$f^{-1}(X)+ f^{-1}(Y) \subseteq f^{-1}(X+Y)$$
If $f$ is surjective we can combine these two, and we have
$$f(f^{-1}(X)+ f^{-1}(Y))=f f^{-1}(X)+f f^{-1}(Y)=X+Y$$
But I need this:
$$f^{-1}(X)+ f^{-1}(Y)=f^{-1}(X+Y)$$
given that $f$ is surjective.

I cannot find the proof and I do not know if it is true, can somebody help me to prove this or give a counterexample or give a reference?

If $a\in f^{-1}(X+Y)$ then $f(a) = x+y$, for some $x$ in $X$ and $y$ in $Y$. If $f$ is surjective then $x = f(b)$ and $y = f(c)$, for some $b,c\in A$. If $d = a-b-c$ then $f(d) = f(a) - f(b) - f(c) = 0.$ It follows that $b+d \in f^{-1}(X)$, and then $a = (b+d) + c \in f^{-1}(X)+ f^{-1}(Y)$.

That shows that $ f^{-1}(X+Y) \subseteq f^{-1}(X)+ f^{-1}(Y)$. You have already shown the reverse inequality, so it follows that $ f^{-1}(X+Y) = f^{-1}(X)+ f^{-1}(Y)$.

 

[steenis]

Thank you very much, This is a great help !

 

[steenis]

Doe anyone know references (books or online) in which this kind of "fomulae" of modules (or groups or rings, ...) are listed and maybe proved ?