A complete structural characterization of submonoids S of a polycyclic-by-finite group such that the semigroup algebra K[S] over a field K is right noetherian is obtained. It follows that such algebras are also left noetherian. 2000 Mathematics Subject Classification 16P40, 16S36, 20M25 (primary), 20F22, 20C07, 20M10 (secondary).
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Jespers et al. (2006) studied this question.