TY - JOUR

T1 - Integrality Over Fixed Rings

AU - Quinn, Declan

PY - 1989/10

Y1 - 1989/10

N2 - In this paper we prove that if G is a finite group of automorphisms of a ring R, where the order of G is a unit in R, then R is fully integral of degree m( G ) over the fixed ring RG. Here m is a function of the order of G. This gives a positive answer to a well-known question of S. Montgomery and extends the result of D. S. Passman for abelian group actions. Our theorem can be viewed as a generalization of the Bergman-Isaacs theorem. In fact that result can be obtained as a corollary, albeit with a poorer index of nilpotence. We then briefly consider duality for Hopf algebra actions and conclude by proving an integrality result for an inner action by a finite-dimensional semisimple Hopf algebra.

AB - In this paper we prove that if G is a finite group of automorphisms of a ring R, where the order of G is a unit in R, then R is fully integral of degree m( G ) over the fixed ring RG. Here m is a function of the order of G. This gives a positive answer to a well-known question of S. Montgomery and extends the result of D. S. Passman for abelian group actions. Our theorem can be viewed as a generalization of the Bergman-Isaacs theorem. In fact that result can be obtained as a corollary, albeit with a poorer index of nilpotence. We then briefly consider duality for Hopf algebra actions and conclude by proving an integrality result for an inner action by a finite-dimensional semisimple Hopf algebra.

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U2 - 10.1112/jlms/s2-40.2.206

DO - 10.1112/jlms/s2-40.2.206

M3 - Article

AN - SCOPUS:0000172449

SN - 0024-6107

VL - s2-40

SP - 206

EP - 214

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 2

ER -