  Let $i: A\to R$ be a ring morphism, and $\chi: R\to A$ a right $R$-linear map
with $\chi(\chi(r)s)=\chi(rs)$ and $\chi(1_R)=1_A$. If $R$ is a Frobenius
$A$-ring, then we can define a trace map $\tr: A\to A^R$. 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 Garc\'{\i}a and del
R\'{\i}o and by D\v{a}sc\v{a}lescu, 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.
