In the ring-shaped tunnel-junction array with four islands, the secondary Coulomb blockade gap in a low bias-voltage range is observed in the I-V characteristics. We attribute its appearance to the unique topology of the array which induces up to two electrons to get trapped inside. We have analyzed the formation and destruction of the gap in terms of detailed single-electron tunneling processes. The negative differential resistance behavior when the thermal and quantum fluctuations are present is also studied. [S0163-1829(99)11203-7].