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PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION

Year 2017, Volume: 21 Issue: 21, 39 - 54, 17.01.2017
https://doi.org/10.24330/ieja.295752

Abstract

Let G be a subgroup of the automorphism group of a commutative
ring with identity T. Let R be a subring of T. We show that RG ⊂ T G
is a minimal ring extension whenever R ⊂ T is a minimal extension under
various assumptions. Of the two types of minimal ring extensions, integral
and integrally closed, both of these properties are passed from R ⊂ T to
RG ⊆ T G. An integrally closed minimal ring extension is a flat epimorphic
extension as well as a normal pair. We show that each of these properties also
pass from R ⊂ T to RG ⊆ T G under certain group action. 

References

  • [6] D. E. Dobbs and J. Shapiro, Descent of divisibility properties of integral domains
  • to fixed rings, Houston J. Math., 32(2) (2006), 337-353.
  • [7] D. E. Dobbs and J. Shapiro, Descent of minimal overrings of integrally closed
  • domains to fixed rings, Houston J. Math., 33(1) (2007), 59-82.
  • [8] D. E. Dobbs and J. Shapiro, Transfer of Krull dimension, lying-over, and
  • going-down to the fixed ring, Comm. Algebra, 35(4) (2007), 1227-1247.
  • [9] D. Ferrand and J.-P. Olivier, Homomorphismes minimaux d’anneaux, J. Algebra,
  • 16 (1970), 461-471.
  • [10] M. Fontana, J. A. Huckaba and I. J. Papick, Pr¨ufer Domains, Monographs
  • and Textbooks in Pure and Applied Mathematics, 203, Marcel Dekker, Inc.,
  • New York, 1997.
  • [11] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago,
  • Revised edition, 1974.
  • [12] M. Knebusch and D. Zhang, Manis Valuations and Pr¨ufer Extensions I, Lecture
  • Notes in Mathematics, 1791, Springer-Verlag, Berlin, 2002.
  • [13] M. E. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc., 20
  • (1969), 193-198
  • [14] G. Picavet and M. Picavet-L’Hermitte, Multiplicative Ideal Theory in Commutative
  • Algebra, Chapter About Minimal Morphisms, 369-386, Springer, New
  • York, 2006.
  • [15] B. Stenstr¨om, Rings of Quotients, Springer-Verlag, New York, Heidelberg
  • Berlin, 1975.
Year 2017, Volume: 21 Issue: 21, 39 - 54, 17.01.2017
https://doi.org/10.24330/ieja.295752

Abstract

References

  • [6] D. E. Dobbs and J. Shapiro, Descent of divisibility properties of integral domains
  • to fixed rings, Houston J. Math., 32(2) (2006), 337-353.
  • [7] D. E. Dobbs and J. Shapiro, Descent of minimal overrings of integrally closed
  • domains to fixed rings, Houston J. Math., 33(1) (2007), 59-82.
  • [8] D. E. Dobbs and J. Shapiro, Transfer of Krull dimension, lying-over, and
  • going-down to the fixed ring, Comm. Algebra, 35(4) (2007), 1227-1247.
  • [9] D. Ferrand and J.-P. Olivier, Homomorphismes minimaux d’anneaux, J. Algebra,
  • 16 (1970), 461-471.
  • [10] M. Fontana, J. A. Huckaba and I. J. Papick, Pr¨ufer Domains, Monographs
  • and Textbooks in Pure and Applied Mathematics, 203, Marcel Dekker, Inc.,
  • New York, 1997.
  • [11] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago,
  • Revised edition, 1974.
  • [12] M. Knebusch and D. Zhang, Manis Valuations and Pr¨ufer Extensions I, Lecture
  • Notes in Mathematics, 1791, Springer-Verlag, Berlin, 2002.
  • [13] M. E. Manis, Valuations on a commutative ring, Proc. Amer. Math. Soc., 20
  • (1969), 193-198
  • [14] G. Picavet and M. Picavet-L’Hermitte, Multiplicative Ideal Theory in Commutative
  • Algebra, Chapter About Minimal Morphisms, 369-386, Springer, New
  • York, 2006.
  • [15] B. Stenstr¨om, Rings of Quotients, Springer-Verlag, New York, Heidelberg
  • Berlin, 1975.
There are 22 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

Amy Schmidt This is me

Publication Date January 17, 2017
Published in Issue Year 2017 Volume: 21 Issue: 21

Cite

APA Schmidt, A. (2017). PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION. International Electronic Journal of Algebra, 21(21), 39-54. https://doi.org/10.24330/ieja.295752
AMA Schmidt A. PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION. IEJA. January 2017;21(21):39-54. doi:10.24330/ieja.295752
Chicago Schmidt, Amy. “PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION”. International Electronic Journal of Algebra 21, no. 21 (January 2017): 39-54. https://doi.org/10.24330/ieja.295752.
EndNote Schmidt A (January 1, 2017) PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION. International Electronic Journal of Algebra 21 21 39–54.
IEEE A. Schmidt, “PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION”, IEJA, vol. 21, no. 21, pp. 39–54, 2017, doi: 10.24330/ieja.295752.
ISNAD Schmidt, Amy. “PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION”. International Electronic Journal of Algebra 21/21 (January 2017), 39-54. https://doi.org/10.24330/ieja.295752.
JAMA Schmidt A. PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION. IEJA. 2017;21:39–54.
MLA Schmidt, Amy. “PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION”. International Electronic Journal of Algebra, vol. 21, no. 21, 2017, pp. 39-54, doi:10.24330/ieja.295752.
Vancouver Schmidt A. PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION. IEJA. 2017;21(21):39-54.