nLab
associated graded ring

Definition

A filtered ring is a filtered object in the category Ring of rings.

The associated graded ring to a filtered ring is the corresponding associated graded object.

Properties

A version of PBW theorem states that if a Lie algebra $g$ over a field $k$ is flat as a $k$-module over a commutative ground ring $k\supset \mathbb{Q}$ containing rationals, then the associated graded ring $\mathrm{Gr}U(g)$ is isomorphic to the symmetric algebra $\mathrm{Sym}(g)$ of the underlying $k$-module of $g$.