### Context

#### Algebra

higher algebra

universal algebra

# Contents

## 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 $Gr U(g)$ is isomorphic to the symmetric algebra $Sym(g)$ of the underlying $k$-module of $g$.

Revised on October 18, 2012 19:37:43 by Zoran Škoda (161.53.130.104)