Associated graded ring
In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:
![\operatorname{gr}_I R = \oplus_{n=0}^\infty I^n/I^{n+1}](/Images/godic/202412/22/b594efc50d8cd53cf84780d1cf9013651320.png")
.
Similarly, if M is a left R-module, then the associated graded module is the graded module over ![\operatorname{gr}_I R](/Images/godic/202412/22/7a414887956f06bccc0ea100b3f836e11320.png")
:
![\operatorname{gr}_I M = \oplus_0^\infty I^n M/ I^{n+1} M](/Images/godic/202412/22/a86af845df925d4d58d4080f65131d431320.png")
.
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