PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION

Year 2017, Volume: 21 Issue: 21, 39 - 54, 17.01.2017

Amy Schmidt

https://doi.org/10.24330/ieja.295752

Cited By: 2

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. 

Keywords

Fixed ring, ring of invariants

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.