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Rings of formal power series with homeomorphic prime spectra

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Abstract

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(T[[X]])→Spec(R[[X]]) 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(T[X])→Spec(R[X]) is a homeomorphism if and only ifR/M⊂T/M is a purely inseparable (algebraic) field extension, whereM is the maximal ideal ofR.

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, 37996-1300, Knoxville, Tennesee, U.S.A.

    David E. Dobbs

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  1. David E. Dobbs
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Dobbs, D.E. Rings of formal power series with homeomorphic prime spectra. Rend. Circ. Mat. Palermo 41, 55–61 (1992). https://doi.org/10.1007/BF02844462

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  • DOI: https://doi.org/10.1007/BF02844462

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