We investigate the effect of a ring resonator on the linewidth and output spectrum of monolithically integrated extended cavity multi-section DBR lasers with an intra-cavity ring resonator. The goal is to achieve an understanding of whether and how the use of an additional ring filter improves the performance of a DBR laser on the aspects of the SMSR and intrinsic linewidth using the capabilities of the InP active-passive integration platform. The laser output spectrum is in good agreement with our theoretical calculations from a steady-state spectral model. A side-mode suppression ratio between 60 and 70 dB is measured for a range of operating semiconductor optical amplifier currents. The frequency noise power spectral density is measured for a range of output power levels. A minimum intrinsic linewidth of 63 kHz is reported. We compare the measured Lorentzian linewidths with our theoretical expectations and present estimates of the possible linewidth improvement with the available photonic integration technology used in this work.