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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!

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    $\begingroup$ This is true for any finite direct sum presentations of $M$ and $N$, irrespective of whether the summands are indecomposable or not. Essentially, direct sum decompositions mean that any homomorphism $\phi$ can be represented by a block matrix of the homomorphisms $\phi_{ij}$. $\endgroup$
    – Igor Khavkine
    Commented Oct 24, 2020 at 19:03
  • $\begingroup$ Thank you Igor. I thought I had seen a statement of this nature before, but couldn't find it written anywhere so I wanted to be sure I had the details right; I'm glad I asked, as I didn't know it was a more general statement! $\endgroup$
    – SCM
    Commented Oct 24, 2020 at 20:55

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