(1) If $M,N,P,Q$ are $R$-modules and $M = P \oplus Q$ then is $M\cap N = (P \cap N) \oplus (Q \cap N)?$
Since the modules are not said to be anything like finitely generated or free, I really don't have any idea regarding this one .
(2) So if this expression can be proved to be false is it true when $M$ is said to be free or finitely generated?
Thanks in advance for help!