  We report the study of far-IR sizes of submillimeter galaxies (SMGs) in
relation to their dust-obscured star formation rate (SFR) and active galactic
nuclei (AGN) presence, determined using mid-IR photometry. We determined the
millimeter-wave ($\lambda_{\rm obs}=1100 \mu$m) sizes of 69 ALMA-identified
SMGs, selected with $\geq10$$\sigma$ confidence on ALMA images ($F_{\rm 1100
\mu m}=1.7$--7.4 mJy). We found that all the SMGs are located above an
avoidance region in the millimeter size-flux plane, as expected by the
Eddington limit for star formation. In order to understand what drives the
different millimeter-wave sizes in SMGs, we investigated the relation between
millimeter-wave size and AGN fraction for 25 of our SMGs at $z=1$--3. We found
that the SMGs for which the mid-IR emission is dominated by star formation or
AGN have extended millimeter-sizes, with respective median $R_{\rm c,e} =
1.6^{+0.34}_{-0.21}$ and 1.5$^{+0.93}_{-0.24}$ kpc. Instead, the SMGs for which
the mid-IR emission corresponds to star-forming/AGN composites have more
compact millimeter-wave sizes, with median $R_{\rm c,e}=1.0^{+0.20}_{-0.20}$
kpc. The relation between millimeter-wave size and AGN fraction suggests that
this size may be related to the evolutionary stage of the SMG. The very compact
sizes for composite star-forming/AGN systems could be explained by supermassive
black holes growing rapidly during the SMG coalescing, star-formation phase.
