Associated graded ring

Summary

The associated graded ring of a commutative ring $R$ with respect to an ideal $I$ is the graded ring $\operatorname{gr}_I(R) = \bigoplus_{n \ge 0} I^n/I^{n+1}$. It keeps the leading terms of elements and forgets the rest. For a local ring and its maximal ideal it describes the tangent cone of the point. Since it is graded, tools for graded rings such as the Hilbert polynomial apply to local rings through it.