Rings of formal power series with homeomorphic prime spectra

LetR?T be domains, not fields, such that Spec(R)=Spec(T) as sets; that is, such that the prime ideals ofT coincide, as sets, with those ofR. It is proved that the canonical map Spec(TX]])→Spec(RX]]) is a homeomorphism. This generalizes a result of Girolami in caseR is a pseudovaluation domain with the SFT (strong finite type)—property andT is its associated valuation domain. The analogous property for polynomial rings is also characterized: Spec(TX])→Spec(RX]) is a homeomorphism if and only ifR/M?T/M is a purely inseparable (algebraic) field extension, whereM is the maximal ideal ofR.