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Some repeating fast radio burst (FRB) sources exhibit complex polarizationbehaviors, including frequency-dependent depolarization, variation of rotationmeasure (RM), and oscillating spectral structures of polarized components. Veryrecently, Feng et al. (2022) reported that active repeaters exhibit conspicuousfrequency-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 welldescribed by multi-path propagation through a magnetized inhomogeneous plasmascreen. This observation strongly suggests that the temporal scattering and RMscatter originate from the same region. Besides, a particular finding of notein 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 sometheoretical predictions on the relations among temporal scattering,depolarization by RM scatter, and PRS contributed by the magnetized plasmaenvironment close to a repeating FRB source. The behaviors of the RM scatterimply that the magnetized plasma environment is consistent with a supernovaremnant or a pulsar wind nebula, and the predicted $\sigma_{\rm RM}$-$\tau_{\rms}$ relation is $\sigma_{\rm RM}\propto\tau_{\rm s}^{(0.54-0.83)}$ fordifferent astrophysical scenarios. We further make a general discussion on PRSthat does not depend on specific astrophysical scenarios. We show that thespecific luminosity of a PRS should have a positive correlation with the RMcontributed by the plasma screen. This is consistent with the observations ofFRB 121102 and FRB 190520B.

Temporal Scattering, Depolarization, and Persistent Radio Emission from Magnetized Inhomogeneous Environments Near Repeating Fast Radio Burst Sources