LetRbe a commutative ring andH Rthe ring of Hurwitz series overR. In this note, we consider some properties of rings, which are shared byRandHR. In particular, we show that for the ringsRandH R, if either ring is (i) a Hermite ring, or (ii) a PF-ring in the sense that every finitely generated projectiveR-module is free, then so is another. We also show that ifRis a PS-ring in the sense that the socleSoc(RR) is projective and char(R) = 0, thenH Ris also a PS-ring.