# Krull-Schmidt Analogue for Complete / Graded Rings

Over the ring $\mathbb{Z}$, all finitely-generated modules decompose uniquely as a direct sum of indecomposable submodules; that's the Krull-Schmidt theorem.

I'm given to understand that if a (commutative, Noetherian) ring $R$ is $\mathbb{N}$-graded over a field $k$ (and the degree zero part of $R$ is equal to $k$), then $R$ satisfies this same Krull-Schmidt condition. I'm told that the same holds if $R$ is a complete local ring over a field.

On the other hand, I can't find a good reference for either of these facts. (I can find references to the statements of both facts, but I dislike the notion of citing an unsupported assertion...) So: can anyone point me to a good proof of a Krull-Schmidt theorem for graded or complete local rings? Thanks!

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Dear Georges, can you say a few more words on how to deduce the graded ring case from Atiyah's article? I was only able to link it to the category of graded modules over $R$. –  Hailong Dao Oct 4 '11 at 15:08