  Radio AGN feedback in X-ray cool cores has been proposed as a crucial
ingredient in the evolution of baryonic structures. However, it has long been
known that strong radio AGNs also exist in "noncool core" clusters, which
brings up the question whether an X-ray cool core is always required for radio
feedback. We present a systematic analysis of 152 groups and clusters to show
that every BCG with a strong radio AGN has an X-ray cool core. Those strong
radio AGNs in the center of the "noncool core" systems identified before are in
fact associated with small X-ray cool cores with typical radii of < 5 kpc (we
call them coronae). Small coronae are most likely of ISM origin and they carry
enough fuel to power radio AGNs. Our results suggest that the traditional cool
core/noncool core dichotomy is too simple. A better alternative is the cool
core distribution function with the enclosed X-ray luminosity. Other
implications of our results are also discussed, including a warning on the
simple extrapolation of the density profile to derive Bondi accretion rate.
