This paper investigates the information theoretic limit of a reconfigurable intelligent surface (RIS) aided communication scenario in which the RIS and the transmitter either jointly or independently send information to the receiver. The RIS is an emerging technology that uses a large number of passive reflective elements with adjustable phases to intelligently reflect the transmit signal to the intended receiver. While most previous studies of the RIS focus on its ability to beamform and to boost the received signal-to-noise ratio (SNR), this paper shows that if the information data stream is also available at the RIS and can be modulated through the adjustable phases at the RIS, significant improvement in the {degree-of-freedom} (DoF) of the overall channel is possible. For example, for an RIS system in which the signals are reflected from a transmitter with $M$ antennas to a receiver with $K$ antennas through an RIS with $N$ reflective elements, assuming no direct path between the transmitter and the receiver, joint transmission of the transmitter and the RIS can achieve a DoF of $\min\left(M+\frac{N}{2}-\frac{1}{2},N,K\right)$ as compared to the DoF of $\min(M,K)$ for the conventional multiple-input multiple-output (MIMO) channel. This result is obtained by establishing a connection between the RIS system and the MIMO channel with phase noise and by using results for characterizing the information dimension under projection. The result is further extended to the case with a direct path between the transmitter and the receiver, and also to the multiple access scenario, in which the transmitter and the RIS send independent information. Finally, this paper proposes a symbol-level precoding approach for modulating data through the phases of the RIS, and provides numerical simulation results to verify the theoretical DoF results.