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Overrings and PM-Spectra

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Part of the Lecture Notes in Mathematics book series (LNM,volume 2103)

Abstract

We aim at analyzing a Prüfer extension AR in terms of the set S(RA) of nontrivial PM-valuations v on R over A (i.e. with AA v ), called the restricted PM-spectrum of R over A, in order to understand the lattice of overrings of A in R. We engage S(RA) as partially ordered set (vw iff A v A w ), although it would be more comprehensive to view S(RA) as a topological space, namely as subspace of the valuation spectrum Spv(R), equipped with one of its well established spectral topologies, whose specialization relation restricts to the partial ordering above. We exhibit several types of Prüfer extensions, where the poset viewpoint has seizable success.

Keywords

  • Overrings
  • Seizable Success
  • Complete Irreducibility
  • Ring Extension
  • Convex Subgroup

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 1.

    This means Proposition 5.9 in Chap. 1. If we cite this result in its own section we just write Proposition 9. If we cite this result in another chapter we write Proposition 1.5.9.

  2. 2.

    Formally this definition makes sense for an arbitrary ring extension, but the set Z(BA) will be useful only in the case that A is ws in B.

  3. 3.

    There are various topologies on this set which give spectral spaces relevant for applications. Here Spv(R) means the same as in [HK].

  4. 4.

    Recall [Vol. I, Definition 3 in II §7].

  5. 5.

    Recall the notations Q(A), M(A), P(A) from [Vol. I, Chap. I §4 & §5].

  6. 6.

    The letter P stands for “Prüfer”, CR stands for “completely reducible”.

  7. 7.

    Recall that P(A) denotes the Prüfer hull of A, cf. [Vol. I, Definition 3 in I §5].

  8. 8.

    Recall that ω(R∕A) denotes the set of minimal elements in the poset S(R∕A).

References

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Knebusch, M., Kaiser, T. (2014). Overrings and PM-Spectra. In: Manis Valuations and Prüfer Extensions II. Lecture Notes in Mathematics, vol 2103. Springer, Cham. https://doi.org/10.1007/978-3-319-03212-2_1

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