Let $M$ be left $R$-module and $A$ ,$B$ are two submodule of $M$ such that $A\times B\cong M$.
Is there submodule $C$ such that $A+C=M$ and $A\cap C=0$?
All my attempts proving the $A$ has a complement in $M$ but it seems to be wrong, but can not find any counter example(s) for it.