I am trying to prove the equivalence of the following statements regarding Module $M_R$. I have no idea about the mechanism of proof in such statements:
Let $A, L\in M_R, \alpha:A \rightarrow L, B \hookrightarrow A, C \hookrightarrow A, M \hookrightarrow L, N \hookrightarrow L$.
Prove: The following statements are equivalent:
(1) $\alpha^{-1}(M+N)=\alpha^{-1}(M)+\alpha^{-1}(N)$.
(2) $(M \cap Im(\alpha))+(N \cap Im(\alpha))=((M+N) \cap Im(\alpha))$.
(3) $(M + Im(\alpha))\cap (N + Im(\alpha))=((M \cap N) + Im(\alpha))$.
I hope you can help with proof or point to a source that helps prove these statements.
Thank you very much