- #1
steenis
- 312
- 18
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?
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?