  We theoretically analyze the effect of electron-electron interactions on
Aharonov-Bohm (AB) current oscillations in ring-shaped systems with metallic
quantum dots pierced by external magnetic field. We demonstrate that
electron-electron interactions suppress the amplitude of AB oscillations
$I_{AB}$ at all temperatures down to T=0 and formulate quantitative predictions
which can be verified in future experiments. We argue that the main physical
reason for such interaction-induced suppression of $I_{AB}$ is electron
dephasing while Coulomb blockade effects remain insignificant in the case of
metallic quantum dots considered here. We also emphasize a direct relation
between our results and the so-called $P(E)$-theory describing tunneling of
interacting electrons.
