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.

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in math and tagged , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s