I am self studying ring theory and modules from Rotman's Advanced Modern Algebra.
I would like some help on putting this thought to bed.
Let $A$ and $B$ be rings. Let $R=A\times B$. Is it possible for $R$-submodules of $R$ to be $A$-submodules of $A$ as well as $B$-submodules of $B$? If yes, I would like to see a proof or a guide to a proof. If not, I'd like to know why?
Thanks.