Within the framework of our program of assessment of the nature of unidentified or poorly known INTEGRAL sources, we present here spectroscopy of optical objects, selected through positional cross-correlation with soft X-ray detections (afforded with satellites such as Swift, ROSAT, Chandra and/or XMM-Newton) as putative counterparts of hard X-ray sources detected with the IBIS instrument onboard INTEGRAL. Using 6 telescopes of various sizes and archival data from two on-line spectroscopic surveys we are able to identify, either for the first time or independent of other groups, the nature of 20 INTEGRAL hard X-ray sources. Our results indicate that: 11 of these objects are active galactic nuclei (AGNs) at redshifts between 0.014 and 0.978, 7 of which display broad emission lines, 2 show narrow emission lines only, and 2 have unremarkable or no emission lines (thus are likely Compton thick AGNs); 5 are cataclysmic variables (CVs), 4 of which are (possibly magnetic) dwarf novae and one is a symbiotic star; and 4 are Galactic X-ray binaries (3 with high-mass companions and one with a low-mass secondary). It is thus again found that the majority of these sources are AGNs or magnetic CVs, confirming our previous findings. When possible, the main physical parameters for these hard X-ray sources are also computed using the multiwavelength information available in the literature. These identifications support the importance of INTEGRAL in the study of the hard X-ray spectrum of all classes of X-ray emitting objects, and the effectiveness of a strategy of multi-catalogue cross-correlation plus optical spectroscopy to securely pinpoint the actual nature of unidentified hard X-ray sources.
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