Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently, $M$ admits a direct sum decomposition into finitely many indecomposable modules, but my question is: When is such an indecomposable decomposition unique?
The case when $M$ is graded seems to have been answered at Krull-Schmidt Analogue for Complete / Graded Rings , but I am interested in the case when $M$ may not be graded.
\oplusis for a binary direct sum, like $R_0 \oplus R_1$. For an $n$-ary direct sum,\bigoplusis more appropriate: compare $\bigoplus_{i \ge 0} R_i$\bigoplus_{i \ge 0} R_ito $\oplus_{i \ge 0} R_i$\oplus_{i \ge 0} R_i, and especially $\displaystyle\bigoplus_{i \ge 0} R_i$\displaystyle\bigoplus_{i \ge 0} R_ito $\displaystyle\oplus_{i \ge 0} R_i$\displaystyle\oplus_{i \ge 0} R_i. I have edited accordingly. $\endgroup$