Global dimensions of power series rings

The present paper aims at seeking the Hilbert syzygies' theorem for the power series ring R[[t]]R[[t]] with respect to the global dimension as well as the weak global dimension. Actually, our results generalize known theorems of Small (1968) [13] and Jondrup and Small (1974) [7] related to these two global dimensions. As a prelude to this, we first transfer some well established results on modules over polynomial rings to modules over power series rings. Also, we prove equivalence of the coherence property of a ring R   with the ℵ0ℵ0-coherence of R   along with flatness of the power series ring R[[t]]R[[t]].