PROPERTIES OF RING EXTENSIONS INVARIANT UNDER GROUP ACTION

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,

PDF

___

[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.

International Electronic Journal of Algebra-Cover

ISSN: 1306-6048
Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2007
Yayıncı: Abdullah HARMANCI

Arşiv