I am studying $\mathbb{N}^n$ graded, finitely generated modules $M$ over the $\mathbb{N}^n$ graded polynomial ring $K[X_1,X_2,X_3...,X_n]$. I know every such $M$ has an indecomposable decomposition $M\cong\bigoplus_{i}I_i$.

Suppose $M\cong \bigoplus_{i}I_i$ and $N\cong \bigoplus_{j}J_j$ are finitely generated modules, and I have a homomorphism $\phi:M\rightarrow N$. My question is: Is it true that $\phi$ is equivalent to a set of homomorphisms $\phi_{ij}:I_i\rightarrow J_j$? More informally, I'm trying to see if I can I study $\phi$ by merely studying how $\phi$ maps between indecomposables.

Thanks!

I am studying $\mathbb{N}^n$-graded, finitely generated modules $M$ over the $\mathbb{N}^n$-graded polynomial ring $K[X_1,X_2,X_3...,X_n]$. I know every such $M$ has an indecomposable decomposition $M\cong\bigoplus_{i}I_i$.

Suppose $M\cong \bigoplus_{i}I_i$ and $N\cong \bigoplus_{j}J_j$ are finitely generated modules, and I have a homomorphism $\phi:M\rightarrow N$. My question is: Is it true that $\phi$ is equivalent to a set of homomorphisms $\phi_{ij}:I_i\rightarrow J_j$? More informally, I'm trying to see if I can I study $\phi$ by merely studying how $\phi$ maps between indecomposables.

Thanks!