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On a single-input-single-out (SISO) interference channel (IC), conventional non-cooperative strategies encourage players selfishly maximizing their transmit data rates, neglecting the deficit of performance caused by and to other players. In the case of proper complex Gaussian noise, the maximum entropy theorem shows that the best-response strategy is to transmit with proper signals (symmetric complex Gaussian symbols). However, such equilibrium leads to degrees-of-freedom zero due to the saturation of interference. With improper signals (asymmetric complex Gaussian symbols), an extra freedom of optimization is available. In this paper, we study the impact of improper signaling on the 2-user SISO IC. We explore the achievable rate region with non-cooperative strategies by computing a Nash equilibrium of a non-cooperative game with improper signaling. Then, assuming cooperation between players, we study the achievable rate region of improper signals. We propose the usage of improper rank one signals for their simplicity and ease of implementation. Despite their simplicity, rank one signals achieve close to optimal sum rate compared to full rank improper signals. We characterize the Pareto boundary, the outer-boundary of the achievable rate region, of improper rank one signals with a single real-valued parameter; we compute the closed-form solution of the Pareto boundary with the non-zero-forcing strategies, the maximum sum rate point and the max-min fairness solution with zero-forcing strategies. Analysis on the extreme SNR regimes shows that proper signals maximize the wide-band slope of spectral efficiency whereas improper signals optimize the high-SNR power offset.