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For $R \hookrightarrow A$ an associative algebra over a ring $R$ equipped with the structure of an augmented algebra $\epsilon \colon A \to R$, the augmentation ideal is the kernel of $\epsilon$.
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 $R$.
Let $G$ be a discrete group and $R$ a ring. Write $R[G]$ for the group algebra of $G$ over $R$.
Write
for the homomorphism of abelian groups which forms the sum of $R$-coefficients of the formal linear combinations that constitute the group ring
This is called the augmentation map. Its kernel
is the augmentation ideal of $\mathbb{Z}[G]$. (It is often denoted by $I(G)$.
The augmentation ideal is indeed a left and right ideal in $R[G]$.
The $R$-module underlying the augmentation ideal of a group algebra is a free module, free on the set of elements
in $R[G]$.
(For the case $R= \mathbb{Z}$)
As a $\mathbb{Z}[G]$-module, considered with the same generators, the relations are generated by those of the form
Last revised on May 6, 2018 at 14:57:29. See the history of this page for a list of all contributions to it.