Submonoids of polycyclic-by-finite groups and their algebras (Q5944688)

It is known that group algebras over fields of polycyclic-by-finite groups are Noetherian, and there even exists the conjecture that these are the only groups with such a property. With this in mind, the authors describe Noetherian semigroup algebras of submonoids of polycyclic-by-finite groups. It is shown that these algebras are finitely presented and they are Jacobson rings. It is also proved that every prime ideal \(P\) of \(K[S]\) (\(K\) a field, \(S\) a submonoid of a polycyclic-by-finite group) is related to a prime ideal of the group algebra of a subgroup of the quotient group of \(S\). Applications to the Krull dimension, prime spectrum, and irreducible \(K[S]\)-modules are given.