Associated graded ring
In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring: gr I R = ⨁ n = 0 ∞ I n / I n + 1 {\displaystyle \operatorname {gr} _{I}R=\bigoplus _{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 gr I R {\displaystyle \operatorname {gr} _{I}R} : gr I M = ⨁ n = 0 ∞ I n M / I n + 1 M {\displaystyle \operatorname {gr} _{I}M=\bigoplus _{n=0}^{\infty }I^{n}M/I^{n+1}M} .