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.Furthermore,let A and B be finite-dimensional algebras over a field k,and let domdim(AX)stand for the dominant dimension of an A-module X.If BMA is a Frobenius bimodule,then domdim(A)≤domdim(BM)and domdim(B)≤domdim(AHomB(M,B)).In particular,if B ? A is a left-split(or right-split)Frobenius extension,then domdim(A)= domdim(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.We also prove that the universal(quantised)enveloping algebras of semisimple Lie algebras aure QF-3 rings in the sense of Morita.