Suppose $f:A\rightarrow B$ is homomorphism of $R-$modules $A$ and $B$. Prove or disprove: if $B_{1}$ is $R-$submodule of $B$, then $f^{-1}(B_{1})$ is $R-$submodule of $A$.
My proof: We know that $f^{-1}(B)$ is subgroup of $A$. Still needs to be shown: $rx\in f^{-1}(B_{1})$, $\forall r \in R, x \in f^{-1}(B_{1})$. Since it is $x \in f^{-1}(B_{1})$ then $f(x) \in B_{1}$. Then we have: $f(rx)=rf(x) \in B_{1}$ from which it follows $rx \in f^{-1}(B_{1})$. I hope the proof is correct.