  We establish relations between Frobenius parts and between flat-dominant
dimensions of algebras linked by Frobenius bimodules. This is motivated by the
Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten
conjecture on stable equivalences. We show that the Frobenius parts of
Frobenius extensions are again Frobenius extensions. Further, let $A$ and $B$
be finite-dimensional algebras over a field $k$, and let $\dm(_AX)$ stand for
the dominant dimension of an $A$-module $X$. If $_BM_A$ is a Frobenius
bimodule, then $\dm(A)\le \dm(_BM)$ and $\dm(B)\le \dm(_A\Hom_B(M, B))$. In
particular, if $B\subseteq A$ is a left-split (or right-split) Frobenius
extension, then $\dm(A)=\dm(B)$. These results are applied to calculate
flat-dominant dimensions of a number of algebras: shew group algebras, stably
equivalent algebras, trivial extensions and Markov extensions. Finally, we
prove that the universal (quantised) enveloping algebras of semisimple Lie
algebras are $QF$-$3$ rings in the sense of Morita.
