By Eric Jespers

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**Extra resources for Non-commutative algebra**

**Sample text**

Promislow) Let G be the group G = x, y | x−1 y 2 x = y −2 , y −1 x2 y = x−2 . Then G is a torsion-free abelian-by-finite group which does not have the unique product property. Proof. Set z = xy, a = x2 , b = y 2 and c = z 2 . Then D = a, b, c is abelian and az = a−1 and bz = b−1 .

Thus KG is a H-graded ring with homogeneous components (KN )h, h ∈ H. Since each h is invertible we get that indeed KG = (KN ) ∗ H, a crossed product. Thus the lemma relates problems of KG to problems of group algebras of subgroups and crossed products. Recall that a group G is polycylic-by-finite if it has a sugroup of finite index that is polycyclic. If {1} = G0 G1 · · · Gn = G is a subnormal series with Gi+1 /Gi either finite or cyclic, then {1} G1 ∩ H · · · Gn ∩ H = H is a subnormal series of H and Gi+1 ∩ H/Gi ∩ H ∼ = (H ∩ Gi+1 )Gi /Gi ⊆ Gi+1 /Gi , so these factors are also finite or cyclic.

Gilmer proved the following result. 14. Let S be an abelian monoid and K a field. Then the monoid algebra K[S] is Noetherian if and only if S is finitely generated. In case K[S] is left and right Noetherian then Okninski showed that Problem 4 also has an affirmative answer. In the case of submonoids S of polycyclic-by-finite groups one can completely characterize when the semigroup algebra K[S] is left and right Noetherian. This result was obtained recently by Jespers and Okninski. 15. Let S be a submonoid of a polycyclic-by-finite group.