Associated graded ring

In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:

${\displaystyle \operatorname {gr} _{I}R=\oplus _{n=0}^{\infty }I^{n}/I^{n+1}}$.

Similarly, if M is a left R-module, then the associated graded module is the graded module over ${\displaystyle \operatorname {gr} _{I}R}$:

${\displaystyle \operatorname {gr} _{I}M=\oplus _{n=0}^{\infty }I^{n}M/I^{n+1}M}$.