  In this paper, we show that the Peccei-Quinn (PQ) symmetry with a good
quality can be realized in a simple $B-L$ extension of the minimal
supersymmetric standard model. The PQ symmetry is a remnant of the $B-L$ gauge
symmetry at the renormalizable level. Besides, the sufficient quality of the PQ
symmetry is preserved by a non anomalous discrete gauged $R$-symmetry and a
small gravitino mass $m_{3/2}\ll 100$GeV. A viable mass range is
$m_{3/2}=\mathcal{O}(1)$eV which allows a high reheating temperature and many
baryogenesis scenarios typified by the thermal leptogenesis without any
astrophysical and cosmological problems. Such a light gravitino may be tested
in the future 21cm line observations.
