ar X iv : m at h / 05 03 68 6 v 1 [ m at h . R A ] 2 9 M ar 2 00 5 FULLY BOUNDED NOETHERIAN RINGS AND FROBENIUS EXTENSIONS

Let i : A → R be a ring morphism, and χ : R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1 R) = 1 A. If R is a Frobenius A-ring, then we can define a trace map tr : A → A R. If there exists an element of trace 1 in A, then A is right FBN if and only if A R is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring, and the condition on the trace can be replaced by a weaker condition. We recover results of García and del Río and by Dˇascˇalescu, Kelarev and Torrecillas on actions of group and Hopf algebras on FBN rings as special cases. We also obtain applications to extensions of Frobenius algebras, and to Frobenius corings with a grouplike element.