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Dimension and decomposition in modular upper-continuous lattices

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We translate notions and results of decomposition and dimension theories for module categories into the lattice environment. In particular, we translate dimension theory in module categories to complete modular upper-continuous lattices.

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References

  1. Golan S.J.: Decomposition and Dimensions in Module Categories. Marcel Dekker, New York (1977)

    MATH  Google Scholar 

  2. Golan, S.J.: Torsion Theories. Longman Scientific and Technical, New York, 29 (1986)

  3. Golan S.J., Simmons H.: Derivatives, nuclei and dimensions on the frame of torsion theories. Longman Scientific and Technical, New York (1988)

    MATH  Google Scholar 

  4. Goldman O.: Elements of noncommutative arithmetic. J. Algebra 35, 308–341 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Grätzer G.: General Lattice Theory, 2nd edn. Birkhäuser, Basel (1998)

    MATH  Google Scholar 

  6. Johnstone P.: Stone Spaces. Cambridge studies in advanced mathematics, London (1992)

    MATH  Google Scholar 

  7. Mitchler G.: Goldman’s primary decomposition and the tertiary decomposition. J. Algebra 16, 129–137 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  8. Picado J., Pultr A.: Frames and Locales: Topology without Points. Frontiers in Mathematics. Birkhäuser/Springer, Basel (2012)

    Book  MATH  Google Scholar 

  9. Popescu, N.: Abelian categories with applications to rings and modules. London Mathematical Society Monographs, vol. 3. Academic Press, New York, (1973)

  10. Simmons H.: Torsion theoretic points and spaces. Proc. Roy. Soc. Edinburgh Sect. A 96, 345–361 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Simmons H.: Near Discreteness of modules and spaces measured by Gabriel and Cantor. J. Pure Appl. Algebra 56, 119–162 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Simmons, H.: A collection of notes on frames. http://www.cs.man.ac.uk/~hsimmons/FRAMES/frames

  13. Simmons H.: A decomposition theory for complete modular meet-continuous lattices. Algebra Universalis 64, 349–377 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Simmons, H.: An introduction to idioms. http://www.cs.man.ac.uk/~hsimmons/00-IDSandMODS/

  15. Simmons, H.: Cantor-Bendixson, socle and atomicity. http://www.cs.man.ac.uk/~hsimmons/00-IDSandMODS/ (2)

  16. Simmons, H.: The Gabriel and the Boyle derivatives for a modular idiom. http://www.cs.man.ac.uk/~hsimmons/00-IDSandMODS/ (4)

  17. Simmons, H.: Gabriel, Loewy, and Cantor-Bendixson sometimes agree http://www.cs.man.ac.uk/~hsimmons/00-IDSandMODS/ (9)

  18. Stenström B.: Rings of Quotients: An Introduction to Methods of Ring Theory. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

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Correspondence to Angel Zaldívar Corichi.

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Presented by F. Wehrung.

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Ríos Montes, J., Zaldívar Corichi, A. Dimension and decomposition in modular upper-continuous lattices. Algebra Univers. 76, 33–51 (2016). https://doi.org/10.1007/s00012-016-0390-3

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