Advertisement

Pairs of rings invariant under group action

  • Nabil ZeidiEmail author
Original Paper
  • 4 Downloads

Abstract

Let \(R\subseteq S\) be a ring extension and \(\mathcal {P}\) be a ring-theoretic property. The pair (RS) is said to be a \(\mathcal {P}\)-pair if, T satisfies \(\mathcal {P}\) for each intermediate ring \(R\subseteq T\subseteq S\). Let G be a subgroup of the automorphism group of S such that R is invariant under the action by G. In this paper we investigate in several cases the transfer of a property \(\mathcal {P}\) from the pair (RS) to \((R^G,S^G)\). For instance, if \(\mathcal {P}:=\) Residaully algebraic, LO, INC, and Valuation, we show that each of these properties pass from (RS) to \((R^G,S^G)\). Additional consequences and applications are given.

Keywords

Normal pair Valuation domain LO & INC Treed domain Ring of invariants Group action 

Mathematics Subject Classification

Primary 13B02 13A50 Secondary 13A15 13A18 13B22 

Notes

References

  1. Ayache, A., Ben Nasr, M., Jarboui, N.: PID pairs of rings and maximal non-PID subrings. Math. Z 268(3–4), 635–647 (2011)MathSciNetCrossRefGoogle Scholar
  2. Ayache, A., Jaballah, A.: Residually algebraic pairs of rings. Math. Z 225, 49–65 (1997)MathSciNetCrossRefGoogle Scholar
  3. Ayache, A., Jarboui, N., Massaoud, E.: Pairs of domains where all intermediate domains are treed. Arab. J. Sci. Eng. 36, 933–946 (2011)MathSciNetCrossRefGoogle Scholar
  4. Ben Nasr, M., Jarboui, N.: On maximal non-valuation subrings. Houst. J. Math. 37(1), 47–59 (2011)MathSciNetzbMATHGoogle Scholar
  5. Bergman, G.M.: Groups acting on hereditary rings. Proc. Lond. Math. Soc. 3(23), 70–82 (1971)MathSciNetCrossRefGoogle Scholar
  6. Davis, E.D.: Overrings of commutative rings III: normal pairs. Trans. Am. Math. Soc. 182, 175–185 (1973)MathSciNetzbMATHGoogle Scholar
  7. Dobbs, D.E.: Lying-over pairs of commutative rings. Can. J. Math. 2, 454–475 (1981)MathSciNetCrossRefGoogle Scholar
  8. Dobbs, D.E., Shapiro, J.: Descent of divisibility properties of integral domains to fixed rings. Houst. J. Math. 32, 337–353 (2006)MathSciNetzbMATHGoogle Scholar
  9. Dobbs, D.E., Shapiro, J.: Descent of minimal overrings of integrally closed domains to fixed rings. Houst. J. Math. 33(1), 59 (2007)MathSciNetzbMATHGoogle Scholar
  10. Dobbs, D.E., Shapiro, J.: Transfer of Krull dimension, lying-over, and going-down to the fixed ring. Commun. Algebra 35, 1227–1247 (2007)MathSciNetCrossRefGoogle Scholar
  11. Ferrand, D., Olivier, J.P.: Homomorphismes minimaux d’anneaux. J. Algebra 16, 461–471 (1970)MathSciNetCrossRefGoogle Scholar
  12. Gilmer, R., Heinzer, W.: Finitely generated intermediate rings. J. Pure Appl. Algebra 37, 237–264 (1985)MathSciNetCrossRefGoogle Scholar
  13. Gilmer, R., Hoffmann, J.: A characterization of Prüfer domains in terms of polynomials. Pac. J. Math. 60, 81–85 (1975)CrossRefGoogle Scholar
  14. Glaz, S.: Fixed rings of coherent regular ring. Commun. Algebra 20, 2635–2651 (1992)MathSciNetCrossRefGoogle Scholar
  15. Kaplansky, I.: Commutative Rings, Revised edition. University of Chicago Press, Chicago (1974)zbMATHGoogle Scholar
  16. Knebusch, M., Zhang, D.: Manis Valuations and Prüfer Extensions I. Springer, Berlin (2002)CrossRefGoogle Scholar
  17. Kumar, R., Gaur, A.: On \(\lambda \)-extensions of commutative rings. J. Algebra Appl. 17(4), 1850063 (2018)MathSciNetCrossRefGoogle Scholar
  18. Kumar, R., Gaur, A.: \(\Delta \)-Extension of rings and invariance properties of ring extension under group action. J. Algebra Appl. 17:1850239 (2018)MathSciNetCrossRefGoogle Scholar
  19. Nagarajan, K.R.: Groups acting on Noetherian rings. Nieuw Arch. Wisk. 16, 25–29 (1968)MathSciNetzbMATHGoogle Scholar
  20. Schmidt, A.: Properties of rings and of ring extensions that are invariant under group action, Ph. D. Dissertation. George Mason University, Spring (2015)Google Scholar
  21. Wadsworth, A.: Pairs of domains where all intermediate domains are Noetherian. Trans. Am. Math. Soc 195, 201–211 (1974)MathSciNetCrossRefGoogle Scholar
  22. Zeidi, N.: On strongly affine extensions of commutative rings. Czechoslov. Math. J. (2019).  https://doi.org/10.21136/CMJ.2019.0240-18 CrossRefzbMATHGoogle Scholar

Copyright information

© The Managing Editors 2020

Authors and Affiliations

  1. 1.Faculty of Sciences, Department of MathematicsSfax UniversitySfaxTunisia

Personalised recommendations