Although inner star-forming rings are common in optical images of barred spiral galaxies, observational evidence for the accompanying molecular gas has been scarce. In this paper we present images of molecular inner rings, traced using the CO (1-0) emission line, from the Berkeley-Illinois-Maryland-Association Survey of Nearby Galaxies (BIMA SONG). We detect inner ring CO emission from all five SONG barred galaxies classified as inner ring (type (r)). We also examine the seven SONG barred galaxies classified as inner spiral (type (s)); in one of these, NGC 3627, we find morphological and kinematic evidence for a molecular inner ring. Inner ring galaxies have been classified as such based on optical images, which emphasize recent star formation. We consider the possibility that there may exist inner rings in which star formation efficiency is not enhanced. However, we find that in NGC 3627 the inner ring star formation efficiency is enhanced relative to most other regions in that galaxy. We note that the SONG (r) galaxies have a paucity of CO and H alpha emission interior to the inner ring (except near the nucleus), while NGC 3627 has relatively bright bar CO and H alpha emission; we suggest that galaxies with inner rings such as NGC 3627 may be misclassified if there are significant amounts of gas and star formation in the bar.
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