TITLE: If $A, B, C$ are finitely generated $\mathbb{Z}/{p^n}\mathbb{Z}$-modules such that $A \oplus B \simeq A \oplus C$, then $B \simeq C$
QUESTION [5 upvotes]: A friend that is preparing for an algebra qualifying exam asked me the following question yesterday, but I really have no idea of how to approach the problem.


Let $A, B$ and $C$ be finitely generated  $\mathbb{Z}/{p^n}\mathbb{Z}$-modules (p is a prime number) such that $A \oplus B \simeq A \oplus C$. Prove that $B \simeq C$.


I would really appreciate if someone can provide me some help with this question.
Thank you very much.

REPLY [4 votes]: What is a $\mathbb{Z}/p^n\mathbb{Z}$-module? It's nothing more than an abelian group $G$ with the property that $p^n G=\{0\}$, no? So, if someone phrased this to you as "If $A,B,C$ are f.g. abelian groups (with the property that they are all annihilated by $p^n$) such that $A\oplus B\cong A\oplus C$, then is $B\cong C$?" What would you say? You, of course, need to be careful to note what type of morphisms you're looking at, and some other identification issues, but it will all work out with a little TLC.