Skip to main content
SpringerLink
Log in

Alexandroff Topology of Algebras Over an Integral Domain

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let S be an integral domain with field of fractions F and let A be an F-algebra. An S-subalgebra R of A is called S-nice if R is lying over S and the localization of R with respect to \(S {\setminus } \{ 0 \}\) is A. Let \({\mathbb {S}}\) be the set of all S-nice subalgebras of A. We define a notion of open sets on \({\mathbb {S}}\) which makes this set a \(T_0\)-Alexandroff space. This enables us to study the algebraic structure of \({\mathbb {S}}\) from the point of view of topology. We prove that an irreducible subset of \({\mathbb {S}}\) has a supremum with respect to the specialization order. We present equivalent conditions for an open set of \(\mathbb S\) to be irreducible, and characterize the irreducible components of \({\mathbb {S}}\). We also characterize quasi-compactness of subsets of a \(T_0\)-Alexandroff topological space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexandroff, P.: Diskrete Räume. Mat. Sb. (N.S.) 2, 501–518 (1937)

    MATH  Google Scholar 

  2. Arenas, F.G.: Alexandroff spaces. Acta Math. Univ. Comenianae 1, 17–25 (1999)

    MathSciNet  MATH  Google Scholar 

  3. Dobbs, D.E., Fontana, M.: Kronecker function rings and abstract Riemann surfaces. J. Algebra 99, 263–274 (1986)

    Article  MathSciNet  Google Scholar 

  4. Dobbs, D.E., Fedder, R., Fontana, M.: Abstract Riemann surfaces of integral domains and spectral spaces. Ann. Mat. Pura Appl. 148, 101–115 (1987)

    Article  MathSciNet  Google Scholar 

  5. Dickmann, M., Schwartz, N., Tressl, M.: Spectral Spaces. Cambridge University Press, Cambridge (2019)

    Book  Google Scholar 

  6. Endler, O.: Valuation Theory. Springer, New York (1972)

    Book  Google Scholar 

  7. Finocchiaro, C.A., Fontana, M., Loper, K.A.: The constructible topology on spaces of valuation domains. Trans. Am. Math. Soc. 365, 6199–6216 (2013)

    Article  MathSciNet  Google Scholar 

  8. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains, Encyclopedia of Mathematics, vol. 93. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  9. Herman, G.T.: On topology as applied to image analysis. Comput. Vis. Graph. Image Process 52, 409–415 (1990)

    Article  Google Scholar 

  10. Hochster, M.: Prime ideal structure in commutative rings. Trans. Amer. Math. Soc. 142, 43–60 (1969)

    Article  MathSciNet  Google Scholar 

  11. Huber, R., Knebusch, M.: On valuation spectra. In: Recent advances in real algebraic geometry and quadratic forms: proceedings of the RAGSQUAD year, Berkeley, 1990–1991, Contemp. Math. 155, Amer. Math. Soc. Providence RI (1994), pp. 167–206

  12. Kronheimer, E.H.: The topology of digital images. Topol. Appl. 46, 279–303 (1992)

    Article  MathSciNet  Google Scholar 

  13. Kuhlmann, F.V.: Places of algebraic fields in arbitrary characteristic. Adv. Math. 188, 399–424 (2004)

    Article  MathSciNet  Google Scholar 

  14. Sarussi, S.: Quasi-valuations extending a valuation. J. Algebra 372, 318–364 (2012)

    Article  MathSciNet  Google Scholar 

  15. Sarussi, S.: Quasi-valuations and algebras over valuation domains. Comm. Algebra (2019). https://doi.org/10.1080/00927872.2018.1522322

  16. Sarussi, S.: Extensions of integral domains and quasi-valuations. Comm. Algebra (2019). https://doi.org/10.1080/00927872.2019.1677695

  17. Sarussi, S.: Quasi-valuations—topology and the weak approximation theorem. Valuation theory in interaction, EMS Series of Congress Reports, EMS Publishing House, 2014, pp. 464–473

  18. Sarussi, S.: Maximal covers of chains of prime ideals. Beitr Algebra Geom 58, 483 (2017). https://doi.org/10.1007/s13366-017-0331-0

    Article  MathSciNet  MATH  Google Scholar 

  19. Sarussi, S.: Totally ordered sets and the prime spectra of rings. Comm. Algebra 45(1), 411–419 (2017). https://doi.org/10.1080/00927872.2016.1175583

    Article  MathSciNet  MATH  Google Scholar 

  20. Stone, M.H.: The theory of representation for Boolean algebras. Trans. Am. Math. Soc. 40, 37–111 (1936)

    MathSciNet  MATH  Google Scholar 

  21. Stone, M.H.: Topological representations of distributive lattices and Brouwerian logics. Časopis Pešt. Mat. Fys. 67, 1–25 (1937)

    MATH  Google Scholar 

  22. Zariski, O.: The compactness of the Riemann manifold of an abstract field of algebraic functions. Bull. Am. Math. Soc. 50, 683–691 (1944)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The author would like to thank the referee for his/her careful reading and valuable comments and suggestions, which substantially improved the paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, SCE College, 84100, Be’er Sheva, Israel

    Shai Sarussi

Authors
  1. Shai Sarussi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shai Sarussi.

Additional information

This paper is dedicated to the memory of Professor Rudolf Bergman.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sarussi, S. Alexandroff Topology of Algebras Over an Integral Domain. Mediterr. J. Math. 17, 54 (2020). https://doi.org/10.1007/s00009-020-1502-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-020-1502-z

Mathematics Subject Classification

Search

Navigation