Based on the two-mode traveling wave model, the nonlinear dynamics of a semiconductor ring laser (SRL) subject to phase conjugate cross optical feedback is numerically studied. It is found that the SRL shows low frequency anti-phase oscillation when the phase changes of these two modes are desynchronized, and the oscillation period increases with the increase of feedback time. In contrast, in the case that the phase changes are synchronized, the two-mode output of the SRL can be one-periodic, multiply-periodic and chaotic by changing injection intensity. In addition, the time-delay signature (TDS) of the generated chaotic signal is identified by the autocorrelation function (ACF), and the corresponding bandwidth is calculated by using the power spectrum. It is found that the TDS first decreases and then increases with the increase of feedback strength, however the bandwidth increases approximately linearly, which indicates that by properly adjusting the feedback strength, the TDS of chaotic signals can be well suppressed and these signals can be used for secure communications and random number generation.