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I have started reading about graded rings and modules and filtered rings and modules. For grading at least,I can see the polynomials as a prototype,graded by usual degree. But I can't seem to find any motivation for filtration other than of course being abstract.

So I guess what I am looking for is :

  1. Some examples of filtration and related concepts that one might usually run into.
  2. Some theorems in algebra that require us to use filtration to prove them.
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    $\begingroup$ One of the most important filtration is the so-called $I$-adic filtration where $I$ is an ideal. If $I$ is an ideal of $R$ and $M$ an $R$-module, you have a filtration $\{I^nM\}$. There are many theorems which use such filtrations, one of the very useful but technical one is called Artin-Rees, for finitely generated modules over a Noetherian ring. Though this is a theorem about filtrations, its use extends far and wide. $\endgroup$ – Mohan Mar 29 at 16:46

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