  We present a study of X-ray AGN overdensities in 16 Abell clusters, within
the redshift range 0.073<z<0.279, in order to investigate the effect of the hot
inter-cluster environment on the triggering of the AGN phenomenon. The X-ray
AGN overdensities, with respect to the field expectations, were estimated for
sources with L_x>= 10^{42} erg s^{-1} (at the redshift of the clusters) and
within an area of 1 h^{-1}_{72} Mpc radius (excluding the core). To investigate
the presence or not of a true enhancement of luminous X-ray AGN in the cluster
area, we also derived the corresponding optical galaxy overdensities, using a
suitable range of $r$-band magnitudes. We always find the latter to be
significantly higher (and only in two cases roughly equal) with respect to the
corresponding X-ray overdensities. Over the whole cluster sample, the mean
X-ray point-source overdensity is a factor of ~4 less than that corresponding
to bright optical galaxies, a difference which is significant at a >0.995
level, as indicated by an appropriate t-student test. We conclude that the
triggering of luminous X-ray AGN in rich clusters is strongly suppressed.
Furthermore, searching for optical SDSS counterparts of all the X-ray sources,
associated with our clusters, we found that about half appear to be background
QSOs, while others are background and foreground AGN or stars. The true
overdensity of X-ray point sources, associated to the clusters, is therefore
even smaller than what our statistical approach revealed.
