  We selected 27244 nearby, red, giant early-type galaxies (RGEs) from the
Sloan Digital Sky Survey (SDSS). In a large fraction (53%) of their spectra the
[O III] emission line is detected, with an equivalent width (EW) distribution
strongly clustered around ~0.75 A. The vast majority of those RGEs for which it
is possible to derive emission line ratios (amounting to about half of the
sample) show values characteristic of LINERs. The close connection between
emission lines and stellar continuum points to stellar processes as the most
likely source of the bulk of the ionizing photons in RGEs, rather than active
nuclei. In particular, the observed EW and optical line ratios are consistent
with the predictions of models in which the photoionization comes from to hot
evolved stars. Shocks driven by supernovae or stellar ejecta might also
contribute to the ionization budget. A minority, ~4%, of the galaxies show
emission lines with an equivalent that is width a factor of ~2 greater than the
sample median. Only among them are Seyfert-like spectra found. Furthermore, 40%
of this subgroup have a radio counterpart, compared to ~6% of the rest of the
sample. These characteristics argue in favor of an AGN origin for their
emission lines. Emission lines diagnostic diagrams do not reveal a distinction
between the AGN subset and the other members of the sample, and consequently
they are not a useful tool for establishing the dominant source of the ionizing
photons, which is better predicted by the EW of the emission lines.
