TITLE: Prove that $\mathbb{Z}/(p^n)$ is indecomposable
QUESTION [5 upvotes]: Could anyone please give me a hint about how to prove that?
I guess I should show that the direct sum is not a cyclic group and get a contradiction but I'm not sure how to start.
thanks

REPLY [4 votes]: Assuming you're talking about $\mathbb Z$-modules, your guess is right.
Let $M=\mathbb Z/(p^n)$.
If $M = A \oplus B$ is a non-trivial decomposition, then $A$ and $B$ are finite groups of order $p^a$ and $p^b$, with $0< a,b <n$. But then, with $c=\max(a,b) < n$, we get that $p^c$ annihilates $M$, which is not true, since $M$ has an element of order $p^n > p^c$.