Let $A, B, C$ be finitely generated modules over PID $F$. It is needed to prove that:
a) $A \oplus B \cong A \oplus C$ implies $B \cong C$
b) If there is an embedding of A into B and vice versa, then $A \cong B$
I believe these should somehow follow from the structure Theorem on finitely generated modules over PIDs, however I am not proficient in algebra, so I have trouble putting the pieces together.