Let $R$ be a principal ideal domain, N,M and P finite generated $R-$modules. If we have $N \oplus P \cong M \oplus P$ how can be proved that $N \cong M$?
My first atempt was to factorise $ 0 \oplus P$ at both sides but if we have a not canonical isomorphism, for example $\phi: N \oplus N \to N \oplus N, (n_1, n_2) \to (n_2, n_1)$ the sugested induced morphism $\bar{\phi}: N \oplus N / 0 \oplus P \to N \oplus N 0 \oplus P$ become a zero morphism. Has anybody an other idea?