# Finitely generated modules over a PID isomorphism

Let $M, N, P$ be finitely generated modules over a PID.

• Show that if $M?M?N?N$, then $M?N$.
• Show that if $M?P?N?P$, then $M?N$.

Answers can be viewed only if
1. The questioner was satisfied and accepted the answer, or
2. The answer was disputed, but the judge evaluated it as 100% correct.
• With $q_{i,\cdot}=q_{i',\cdot}$, just that we are pairing off the two copies of each elementary divisor. And in the second page, yes. If we map one elementary divisor of M to a another in N then we map the corresponding paired one in M to the corresponding paired one in N.