Semigroup algebras of submonoids of polycyclic-by-finite groups and maximal orders

TL;DR

This paper establishes criteria for when certain algebraic structures derived from submonoids of polycyclic-by-finite groups are maximal orders, extending previous results to more general cases based solely on monoid properties.

Contribution

It provides necessary and sufficient conditions for prime Noetherian algebras of submonoids to be maximal orders, generalizing earlier work to broader algebraic settings.

Findings

Criteria for maximal orders in submonoid algebras

Extension of Brown's results to non-PI cases

Conditions depend solely on the monoid S

Abstract

Necessary and sufficient conditions are given for a prime Noetherian algebra K[S] of a submonoid S of a polycyclic-by-finite group G to be a maximal order. These conditions are entirely in terms of the monoid S. This extends earlier results of Brown concerned with the group ring case and of the authors for the case where K[S] satisfies a polynomial identity.