Review question from my first course on Algebraic Structures:
Let $M, N, P$ be finitely generated modules over a PID.
- a) Show that if $M⊕M≅N⊕N$, then $M≅N$.
- b) Show that if $M⊕P≅N⊕P$, then $M≅N$.
My attempt so far (not finished):
a) We know, from the uniqueness of the theorem about elementary divisors, that $M$ and $N$ are isomorph if and only if they have the same rank and the same list of elementary divisors.
And that any finitely generated module over a PID $R$ can be expressed as
$R^r \oplus R/(p_1^{k_1}) \oplus R/(p_2^{k_2}) \oplus \cdots \oplus R/(p_n^{k_n})$
where $p_1^{k_1},\ldots,p_n^{k_n}$ are prime powers in $R$.
Let $m,n$ denote the number of times $R/(p^k)$ appears as an elementary divisor of $M,N$ respectively.
Since $M⊕M≅N⊕N$,
$m+m=n+n$
and then $m=n$
The same argument can be made for the rank of $R^r$.
Am I correct so far? I understand that I still need to show that the list of elementary divisors is the same.
How would I go about doing that? I'm a bit lost after this point.