Augmentation ideal

In algebra, an augmentation ideal is an ideal that can be defined in any group ring. 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 ε {displaystyle varepsilon } , called the augmentation map, from the group ring R [ G ] {displaystyle R} to R, defined by taking a (finite) sum ∑ r i g i {displaystyle sum r_{i}g_{i}} to ∑ r i . {displaystyle sum r_{i}.} (Here r i ∈ R {displaystyle r_{i}in R} and g i ∈ G {displaystyle g_{i}in G} .) In less formal terms, ε ( g ) = 1 R {displaystyle varepsilon (g)=1_{R}} for any element g ∈ G {displaystyle gin G} , ε ( r ) = r {displaystyle varepsilon (r)=r} for any element r ∈ R {displaystyle rin R} , and ε {displaystyle varepsilon } is then extended to a homomorphism of R-modules in the obvious way. The augmentation ideal A is the kernel of ε {displaystyle varepsilon } and is therefore a two-sided ideal in R. A is generated by the differences g − g ′ {displaystyle g-g'} of group elements. Equivalently, it is also generated by { g − 1 : g ∈ G } {displaystyle {g-1:gin G}} , which is a basis as a free R-module. For R and G as above, the group ring R is an example of an augmented R-algebra. Such an algebra comes equipped with a ring homomorphism to R. The kernel of this homomorphism is the augmentation ideal of the algebra. The augmentation ideal plays a basic role in group cohomology, amongst other applications.