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 $M?N$.
Abstract Algebra
Ssvnormandysr1 Ssvnormandysr1
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Dtatdm Dtatdm
  • Ssvnormandysr1 Ssvnormandysr1

    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?

  • Dtatdm Dtatdm

    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.

The answer is accepted.