# nLab augmented algebra

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Definition

For $R$ a ring, an associative algebra over $R$ is a ring $A$ equipped with a ring inclusion $R↪A$. If $A$ is also equipped with a ring homomorphism the other way round,

$ϵ:A\to R$\epsilon \colon A \to R

then it is called an augmented algebra.

The kernel of $ϵ$ is called the corresponding augmentation ideal in $A$.

## Examples

Every group algebra $R\left[G\right]$ is canonically augmented, the augmentation map being the operation that forms the sum of coefficients of the canonical basis elements.

Created on October 14, 2012 17:42:54 by Urs Schreiber (89.204.130.129)