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 $M?N$.
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Hey, just so it's clear, in the 9th line of the first page what does it say after the "with"? Also, 6th line of the second page it says "q_i,M arrow q_j,N" right?

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.
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