Title: A connection between the good property of an Artinian Gorenstein local ring and that of its quotient modulo socle,
Abstract: Following Roos, we say that a local ring $R$ is good if all finitely
generated $R$-modules have rational Poincaré series over $R$, sharing a
common denominator. Rings with the Backelin-Roos property and generalised Golod
rings are good due to results of Levin and Avramov respectively. Let $R$ be an
Artinian Gorenstein local ring. The ring $R$ is shown to have the Backelin-Roos
property if $R/ soc(R)$ is a Golod ring. Furthermore the ring $R$ is
generalised Golod if and only if $R/ soc(R)$ is so.
We explore when connected sums of Artinian Gorenstein local rings are good.
We provide a uniform argument to show that stretched, almost stretched
Gorenstein rings are good and show further that the Auslander-Reiten conjecture
holds true for such rings. We prove that Gorenstein rings of multiplicity at
most eleven are good. We recover a result of Rossi-Şega on the good
property of compressed Gorenstein local rings in a stronger form by a shorter
argument.