Exercise 23, § 4

Let $A$ be a principal ideal domain and let $M$ be a finitely generated $A$-module. Show that if $N$ and $P$ are two $A$-modules such that $M \oplus N$ and $M \oplus P$ are isomorphic, then $N$ and $P$ are isomorphic. (Reduce to the case where $M$ is cyclic ; by identifying $M \oplus N$ with $M \oplus P$, we may consider an $A$-module $E$ decomposed in two ways as a direct sum of submodules $F \oplus N$ and $G \oplus P$, where $F$ and $G$ are cyclic and isomorphic. In the case where $F$ and $G$ are isomorphic to $A$, show that if $N \neq P$ then $N$ is the direct sum of $N \cap P$ and a submodule isomorphic to $A$. If $F$ and $G$ are isomorphic to $A/\mathbf{A}\pi^n$, where $\pi$ is an irreducible element of $A$, consider two generators $a, b$ of $F, G$ respectively, and examine the cases $\pi^n a \notin G,\ \pi^n b \notin F$, and finally the case where both $\pi^{n-1}a \in G$ and $\pi^{n-1}b \in F$; in this last case, consider the element $a + b$ of $E$ and the submodule it generates.)