  Some repeating fast radio burst (FRB) sources exhibit complex polarization
behaviors, including frequency-dependent depolarization, variation of rotation
measure (RM), and oscillating spectral structures of polarized components. Very
recently, Feng et al. (2022) reported that active repeaters exhibit conspicuous
frequency-dependent depolarization and a strong correlation between RM scatter
($\sigma_{\rm RM}$) and the temporal scattering time ($\tau_{\rm s}$),
$\sigma_{\rm RM}\propto\tau_{\rm s}^{1.0\pm0.2}$, both of which can be well
described by multi-path propagation through a magnetized inhomogeneous plasma
screen. This observation strongly suggests that the temporal scattering and RM
scatter originate from the same region. Besides, a particular finding of note
in Feng et al. (2022) is that the FRBs with compact persistent radio sources
(PRS) tend to have extreme $\sigma_{\rm RM}$. In this work, we focus on some
theoretical predictions on the relations among temporal scattering,
depolarization by RM scatter, and PRS contributed by the magnetized plasma
environment close to a repeating FRB source. The behaviors of the RM scatter
imply that the magnetized plasma environment is consistent with a supernova
remnant or a pulsar wind nebula, and the predicted $\sigma_{\rm RM}$-$\tau_{\rm
s}$ relation is $\sigma_{\rm RM}\propto\tau_{\rm s}^{(0.54-0.83)}$ for
different astrophysical scenarios. We further make a general discussion on PRS
that does not depend on specific astrophysical scenarios. We show that the
specific luminosity of a PRS should have a positive correlation with the RM
contributed by the plasma screen. This is consistent with the observations of
FRB 121102 and FRB 190520B.
