Finiteness and homological conditions in commutative group rings
This paper surveys finiteness properties and homological properties for commutative group rings. In the following, \(R\) denotes a commutative ring, \(G\) is an abelian group, and \(RG\) is the associated group ring.NEWLINENEWLINEIn Section 2, a number of finiteness properties for \(RG\) are characterized via finiteness properties for \(R\) and \(G\). For example, \(RG\) is noetherian (respectively, artinian) if and only if \(R\) is noetherian (respectively, artinian) and \(G\) is finitely generated (respectively, finite). Properties such as coherence and UFD are also discussed.NEWLINENEWLINEIn Section 3, some homological properties for \(RG\) are characterized via properties for \(R\) and \(G\). For example \(RG\) is von Neumann regular if and only if \(R\) is von Neumann regular, \(G\) is torsion, and \(R\) is uniquely divisible by the order of every element of \(G\). Properties such as finiteness of the weak global dimension and regularity are also discussed.NEWLINENEWLINESection 4 focuses on certain zero divisor controlling conditions. For example, \(RG\) is an integral domain if and only if \(R\) is an integral domain and \(G\) is torsion free. And if \(G\) is torsion free, then \(RG\) is a PF (respectively PP) ring if and only if \(R\) is a PF (respectively PP) ring.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13006].
scientific article; zbMATH DE number 6052219