  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 AGN also exist in "noncool core" clusters, which brings
up the question whether an X-ray cool core is always required for radio
feedback. In this work, we present a systematic analysis of BCGs and strong
radio AGN in 152 groups and clusters from the Chandra archive. All 69 BCGs with
radio AGN more luminous than 2x10^23 W Hz^-1 at 1.4 GHz are found to have X-ray
cool cores. The BCG cool cores can be divided into two classes, the
large-cool-core (LCC) class and the corona class. Small coronae, easily
overlooked at z>0.1, can trigger strong heating episodes in groups and
clusters, long before large cool cores are formed. Strong radio outbursts
triggered by coronae may destroy embryonic large cool cores and thus provide
another mechanism to prevent formation of large cool cores. However, it is
unclear whether coronae are decoupled from the radio feedback cycles as they
have to be largely immune to strong radio outbursts. Our sample study also
shows the absence of groups with a luminous cool core while hosting a strong
radio AGN, which is not observed in clusters. This points to a greater impact
of radio heating on low-mass systems than clusters. As examples of the corona
class, we also present detailed analyses of a BCG corona associated with a
strong radio AGN (ESO 137-006 in A3627) and one of the faintest coronae known
(NGC 4709 in the Centaurus cluster). 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 or gas
mass.
