Augmentation ideal

In algebra, an augmentation ideal is an ideal that can be defined in any group ring.

If G is a group and R a commutative ring, there is a ring homomorphism ${\displaystyle \varepsilon }$, called the augmentation map, from the group ring ${\displaystyle R[G]}$ to R, defined by taking a (finite) sum ${\displaystyle \sum r_{i}g_{i}}$ to ${\displaystyle \sum r_{i}.}$ (Here r_i∈R and g_i∈G.) In less formal terms, ε(g)=1_R for any element g∈G, ${\displaystyle \varepsilon (r)=r}$ for any element r∈R, and ${\displaystyle \varepsilon }$ is then extended to a homomorphism of R-modules in the obvious way.

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