Following Roos, we say that a local ring $R$ is good if all finitely generated $R$-modules have rational Poincar\'e 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. If $R$ is a Gorensten ring such that $R/ soc(R)$ is a Golod ring, then the ring $R$ is shown to have the Backelin-Roos property. We show that fibre products of local algebras over a field with the Backelin-Roos property also have the same property. We identify a certain quotient $C$ of the Koszul complex of a Gorenstein local ring $R$ such that $R$ has the Backelin-Roos property whenever $C$ is a Golod algebra. It is proved that for a Gorenstein ring $R$, the ring $R$ is generalised Golod if and only if $R/ soc(R)$ is so. We show that fibre products of local rings, connected sums of Gorenstein local rings are generalised Golod if and only if constituent rings are so. We provide a uniform argument to sho 查看全文>>