The properties of the ground state of a closed dot-ring system with a magnetic flux in the Kondo regime are studied theoretically by means of a one-impurity Anderson Hamiltonian. The Hamiltonian is solved by means of the Slave-Boson mean-field theory. It is shown that at T＝０, a suppressed Kondo effect exists in this system even when the mean level spacing of electrons in the ring is larger than the bulk Kondo temperature; the physical quantities of this system depend sensitively on both the parity of the system and the size of the ring; the rich physical behaviors of this system can attribute to the coexistence of both the f inite-size effect and the Kondo screening effect in this system. It is also poss ible to detect the Kondo screening cloud by measuring the persistent current or the zero_field impurity susceptibility χimp directly in future exp eriments.