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augmented algebra

Definition

For $R$ a ring, an associative algebra over $R$ is a ring $A$ equipped with a ring inclusion $R\hookrightarrow A$. 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 $R[G]$ is canonically augmented, the augmentation map being the operation that forms the sum of coefficients of the canonical basis elements.

Related concepts

• augmentation