nLab
augmented algebra

Contents

Definition

For RR a ring, an associative algebra over RR is a ring AA equipped with a ring inclusion RAR \hookrightarrow A.

Definition

If the RR-algebra AA is equipped with a ring homomorphism the other way round,

ϵ:AR \epsilon \colon A \to R

then it is called an augmented algebra.

Remark

In Cartan-Eilenberg this is called a supplemented algebra.

Definition

The kernel of ϵ\epsilon is called the corresponding augmentation ideal in AA.

Examples

Example

An augmentation of a bare ring itself, being an associative algebra over the ring of integers \mathbb{Z}, is a ring homomorphism to the integers

ϵ:R \epsilon \colon R \to \mathbb{Z}
Example

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

References

Revised on June 19, 2013 20:16:34 by Urs Schreiber (82.169.65.155)