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.
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