Ultraproducts of Group Rings
نویسنده
چکیده
Group Rings Let G = g1, g2, . . . , gn be a finite group, and let k be a field. We define the group ring k[G] to be the set of sums of the form a1g1 + a2g2 + · · ·+ angn with each ai ∈ k and gi ∈ G. Addition is defined componentwise, i.e. (a1g1 + a2g2 + · · ·+ angn) + (b1g1 + b2g2 + · · ·+ bngn) = ((a1 + b1)g1 + (a2 + b2)g2 + · · ·+ (an + bn)gn). We define multiplication in the following way: (agi)× (bgj) = (ab)(gigj), where ab ∈ k and gigj ∈ Gi. Given this formula, we can see that multiplication of two elements of the group ring is carried out in the following way: (a1g1 + a2g2 + · · ·+ angn) + (b1g1 + b2g2 + · · ·+ bngn)
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