We investigate persistent currents and scattering phases in one-dimensional rings coupled to external leads, with paying attention to the role of inversion-symmetry and a threading magnetic flux. It is found that for a Aharonov-Bohm ring there exist flux-independent perfect transmission and reflection, leading to giant persistent current when the energy approaches to those singular points. Interestingly, the transmittance oscillation driven by the magnetic flux can also be manifested with related to inversion symmetry broken by different upper-and lower-branch length. In this case, characteristic double-zero points show up in the transmittance, accompanied by 2p-jumps in a transmittance phase. We also examine how time-reversal symmetry affect the transmission properties by considering a system in which both broken inversion-symmetry and a magnetic flux are presented.