nLab
augmentation ideal

Idea

For $R\hookrightarrow A$ an associative algebra over a ring $R$ equipped with the structure of an augmented algebra $ϵ:A\to R$, the augmentation ideal is the kernel of $ϵ$.

Specifically for $G$ a group, and $R[G]$ its group algebra over a ring $R$, the augmentation ideal is the ideal in $R[G]$ which consists of those formal linear combinations over $R$ of elements in $G$ whose sum of coefficients vanishes in $R$.

Examples

For group algebras

Let $G$ be a discrete group and $R$ a ring. Write $R[G]$ for the group algebra of $G$ over $R$.

Definition

Write

$$ϵ:\mathbb{Z}[G]\to \mathbb{Z}$$\epsilon \colon \mathbb{Z}[G] \to \mathbb{Z}

for the homomorphism of abelian groups which forms the sum of $R$-coefficients of the formal linear combinations that constitute the group ring

$$ϵ:r\mapsto \sum _{g\in G}{r}_g.$$\epsilon \colon r \mapsto \sum_{g \in G} r_g \,.

This is called the augmentation map. Its kernel

$$\ker (ϵ)\hookrightarrow \mathbb{Z}[G]$$ker(\epsilon) \hookrightarrow \mathbb{Z}[G]

is the augmentation ideal of $\mathbb{Z}[G]$.

Properties

General

Proposition

The augmentation ideal is indeed a left and right ideal in $R[G]$.

For group algebras

Proposition

The $R$-module underlying the augmentation ideal of a group algebra is a free module, free on the set of elements

$$\{g-e\mid g\in G,g\ne e\}$$\{ g - e | g \in G,\; g \neq e \}

in $R[G]$.