Semigroup Algebras and Noetherian Maximal Orders: a Survey

Abstract

A survey is given on recent results describing when a semigroup algebra K[S] of a submonoid S of a polycyclic-by-finite group is a prime Noetherian maximal order. As an application one constructs concrete classes of finitely presented algebras that have the listed properties. Also some open problems are stated.Keywords Semigroup algebra · Noetherian · Maximal orderIn Sect. 2 we give necessary and sufficient conditions for a semigroup algebra K[S] of a submonoid S of a polycyclic-by-finite group to be a prime Noetherian maximal order. These results show that the problem can be reduced to S. The proofs rely on properties of minimal prime ideals of S and height one prime ideals of K [S]. These properties are being dealt with in Sect. 3. To obtain concrete examples, in Sect. 4 we deal with the Abelian case. These Abelian maximal orders are later used to show that some concrete non-commutative examples are maximal orders as well. Furthermore, in Sect. 5, we investigate a non-commutative class of examples, the monoids of I -type and their algebras. In order to construct such monoids one needs to describe finite solvable groups that act on finite Abelian groups of the same order and that satisfy some extra property. Such groups, called Involutive Yang-Baxter groups, are considered in Sect 6. Finally, in Sect. 7, the concrete class of the monoids of IG-type is studied. This class contains the monoids of I -type and provides many more examples of maximal orders. Furthermore, algebras K[S] of monoids of IG-type turn out to be epimorphic images of algebras K[T ] with T a monoid of I -type.Research partially supported by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Belgium), Flemish-Polish bilateral agreement BIL2005/VUB/06.

Krull orders in nilpotent groups: corrigendum and addendum

“…This then yields a class of non-Noetherian algebras that have a nice arithmetical structure. For more information on Krull orders and Krull monoids, we refer the reader to the monographs [5,8] and to some older and recent papers [1,2,4,6,9].…”