Noetherian Semigroup Algebras
Jan Okniński · 2020
It is well known, and easy to prove, that, for a group G , the algebra K [ G ] is right noetherian if and only if it is left noetherian. This is due to the antiautomorphism of K [ G ] determined by g → g −1 for g ∊ G . Moreover, in this case, G has the a.c.c. on subgroups; see [ 203 ], § 10.2. This seems to be the only general result on noetherian group algebras. On the other hand, a classical result of Higman asserts that the group algebra of a polycyclic-by-finite group is noetherian. A very deep structure, and representation theory of these algebras have recently been developed. For a survey of this topic, we refer to [ 204 ]. This motivated the results in Chapter 11 dealing with subsemigroups of polycyclic-by-finite groups. We note that no known examples of noetherian group algebras arise from groups that are not polycyclic-by-finite.