I'm trying to show that if $R$ is an Artinian ring, then for finitely generated modules $M,N,N'$, we have that $M\oplus N\cong M\oplus N'$ implies that $N\cong N'$.
I'm supposed to do this by decomposing $M,N,N'$ and then using the Krull-Schmidt Theorem.
For the first part, this means writing $M,N,N'$ as direct sums of non-zero submodules. As $R$ is Artinian, finitely generated modules are of finite length (from Hopkins' Theorem) and hence these decompositions will be of finite length. Therefore, we can use the Krull-Schmidt Theorem, which says that these indecomposable components are uniquely determined up to isomorphism.
However, this only gives us information about the components of the decompositions of these modules - not about the relations between them (specifically, between $N$ and $N'$).
How can I use the Krull-Schmidt Theorem to get the desired result?