Chain conditions and semigroup graded rings

Abstract

The following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R [S] we also give a solution to the problem in case S is an inverse semigroup.

Semigroup rings with certain finiteness conditions

“…It was proved in [3] that for rings graded by polycyclic-by-finite groups, the homogeneous Noether property implies the Noether property. Jespers [15] considered the conditions of the homogeneous Noether property of the rings Rs, where S is a certain submonoid of a polycyclic-by-finite group.…”

“…The rapidly growing theory of semigroup-graded rings has not been represented in monographs so far. However, in [15], there are many results concerning the finiteness conditions in these rings. We refer the reader to [14,9,5] for the concepts and facts of the ring and semigroup theories.…”