Full duplex (FD) radio has attracted extensive attention due to its co-time and co-frequency transceiving capability. {However, the potential gain brought by FD radios is closely related to the management of self-interference (SI), which imposes high or even stringent requirements on SI cancellation (SIC) techniques. When the FD deployment evolves into next-generation mobile networking, the SI problem becomes more complicated, significantly limiting its potential gains.} In this paper, we conceive a multi-cell FD networking scheme by deploying a reconfigurable intelligent surface (RIS) at the cell boundary to configure the radio environment proactively. To achieve the full potential of the system, we aim to maximize the sum rate (SR) of multiple cells by jointly optimizing the transmit precoding (TPC) matrices at FD base stations (BSs) and users and the phase shift matrix at RIS. Since the original problem is non-convex, we reformulate and decouple it into a pair of subproblems by utilizing the relationship between the SR and minimum mean square error (MMSE). The optimal solutions of TPC matrices are obtained in closed form, while both complex circle manifold (CCM) and successive convex approximation (SCA) based algorithms are developed to resolve the phase shift matrix suboptimally. Our simulation results show that introducing an RIS into an FD networking system not only improves the overall SR significantly but also enhances the cell edge performance prominently. More importantly, we validate that the RIS deployment with optimized phase shifts can reduce the requirement for SIC and the number of BS antennas, which further reduces the hardware cost and power consumption, especially with a sufficient number of reflecting elements. As a result, the utilization of an RIS enables the originally cumbersome FD networking system to become efficient and practical.