TITLE: $M \oplus M \simeq N \oplus N$ then $M \simeq N.$
QUESTION [7 upvotes]: Let $M$ and $N$ be finitely generated $R$-modules where $R$ principal domain. Show that if $M \oplus M \simeq N \oplus N$ then $M \simeq N.$

REPLY [2 votes]: Another way of putting it is to consider that the isomorphism class of a finitely-generated module $M$ over a PID $R$ is uniquely determined by the sequence of its invariant factors
$$
(a_{1}) \supseteq  (a_{2}) \supseteq \dots \supseteq (a_{k}),
$$
with all $a_{i}$ not units, and 
$$
M \cong \bigoplus_{i=1}^{k} \frac{R}{(a_{i})}.
$$
Clearly the sequence of invariant factors for $M \oplus M$ is just
$$
(a_{1}) \supseteq  (a_{1}) \supseteq  (a_{2}) \supseteq  (a_{2}) \supseteq  \dots \supseteq  (a_{k}) \supseteq  (a_{k}).
$$
Now do the same for $N$ and $N \oplus N$, and compare.