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Following Roos, we say that a local ring $R$ is good if all finitelygenerated $R$-modules have rational Poincar\'e series over $R$, sharing acommon denominator. Rings with the Backelin-Roos property and generalised Golodrings are good due to results of Levin and Avramov respectively. Let $R$ be anArtinian Gorenstein local ring. The ring $R$ is shown to have the Backelin-Roosproperty if $R/ soc(R)$ is a Golod ring. Furthermore the ring $R$ isgeneralised 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 stretchedGorenstein rings are good and show further that the Auslander-Reiten conjectureholds true for such rings. We prove that Gorenstein rings of multiplicity atmost eleven are good. We recover a result of Rossi-\c{S}ega on the goodproperty of compressed Gorenstein local rings in a stronger form by a shorterargument.

A connection between the good property of an Artinian Gorenstein local ring and that of its quotient modulo socle