  We present a study of the nuclear activity in a well-defined sample of the
most isolated galaxies in the local Universe traced by their far-infrared (FIR)
and radio continuum emission. We use the well-known radio continuum-FIR
correlation to select radio-excess galaxies that are candidates to host an
active galactic nucleus (AGN), as well as the FIR colours to find obscured AGN
candidates. The existing information on nuclear activity in the V\'eron-Cetty
catalogue and in the NASA Extragalactic Database are also used. A final
catalogue of AGN-candidate galaxies has been produced. It contains 89 AGN
candidates and is publicly available on the AMIGA web page
(http://www.iaa.csic.es/AMIGA.html). At most ~ 1.5 % of the galaxies shows a
radio-excess with respect to the radio-FIR correlation, and this fraction even
goes down to less than 0.8 % after rejection of back/foreground sources. We
find that the fraction of FIR colour selected AGN-candidates is ~ 28 % with a
lower limit of ~ 7 %. A comparison with the results from the literature shows
that the AMIGA sample has the lowest ratio of AGN candidates, both globally and
separated into early and late types. Field galaxies as well as poor cluster and
group environments show intermediate values, while the highest rates of AGN
candidates are found in the central parts of clusters and in pair/merger
dominated samples. We conclude that the environment plays a crucial and direct
role in triggering radio nuclear activity and not only via the
density-morphology relation. Isolated, early-type galaxies show a particularly
low level of activity at radio wavelengths hence constituting the most
nurture-free population of luminous early-type galaxies.
