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A new interference management scheme based on integer forcing (IF) receivers is studied for the two-user multiple-input and multiple-output (MIMO) interference channel. The proposed scheme employs a message splitting method that divides each data stream into common and private sub-streams, in which the private stream is recovered by the dedicated receiver only while the common stream is required to be recovered by both receivers. Specifically, to enable IF sum decoding at the receiver side, all streams are encoded using the same lattice code. Additionally, the number of common and private streams of each user is carefully determined by considering the number of antennas at transmitters and receivers, the channel matrices, and the effective signal-to-noise ratio (SNR) at each receiver to maximize the achievable rate. Furthermore, we consider various assumptions of channel state information at the transmitter side (CSIT) and propose low-complexity linear transmit beamforming suitable for each CSIT assumption. The achievable sum rate and rate region are analytically derived and extensively evaluated by simulation for various environments, demonstrating that the proposed interference management scheme strictly outperforms the previous benchmark schemes in a wide range of channel parameters due to the gain from IF sum decoding.