In an article I am reading I found the following statement:
If $D$ is a PID, then every finitely generated $D$-module is isomorphic to a direct sum of cyclic $D$-modules. That is, it decomposes uniquely into the form $$ D^\beta\oplus\biggl(\bigoplus_{i=1}^mD/d_iD\biggr)\text, $$ for $d_i\in D$, $\beta\in\Bbb Z$, such that $d_i\mid d_{i+1}$. Similarly, every graded module $M$ over a graded PID $D$ decomposes uniquely into the form $$ \biggl(\bigoplus_{i=1}^n\Sigma^{\alpha_i}D\biggr)\oplus\biggl(\bigoplus_{j=1}^m\Sigma^{\gamma_j}D/d_jD\biggr)\text, $$ where $d_j\in D$ are homogeneus elements so that $d_j\mid d_{j+1}$, $\alpha_i,\gamma_j\in\Bbb Z$, and $\Sigma^\alpha$ denotes an $\alpha$-shift upward in grading.
I have no problem in understanding the first part because it is the standard structure theorem for finitely generated modules over a PID. But I can't understand the second decomposition: what do exactly $\Sigma^{\alpha_i}D$ and $\Sigma^\gamma_jD/d_jD$ mean? Does the second decomposition follow easily from the first or is it an independent theorem? I can't find this decomposition in any other place.