## Structure Theorem for finitely generated (graded) modules over a PID

If $R$ is a PID, then every finitely generated module $M$ over $R$ is isomorphic to a direct sum of cyclic $R$-modules. That is, there is a unique decreasing sequence of proper ideals $(d_1)\supseteq(d_2)\supseteq\dots\supseteq(d_m)$ such that $\displaystyle M\cong R^\beta\oplus\left(\bigoplus_{i=1}^m R/(d_i)\right)$ where $d_i\in R$, and $\beta\in\mathbb{Z}$.

Similarly, every graded module $M$ over a graded PID $R$ decomposes uniquely into the form $\displaystyle M\cong\left(\bigoplus_{i=1}^n\Sigma^{\alpha_i}R\right)\oplus\left(\bigoplus_{j=1}^m\Sigma^{\gamma_j}R/(d_j)\right)$ where $d_j\in R$ are homogenous elements such that $(d_1)\supseteq(d_2)\supseteq\dots\supseteq(d_m)$, $\alpha_i, \gamma_j\in\mathbb{Z}$, and $\Sigma^\alpha$ denotes an $\alpha$-shift upward in grading.