Recently semiconductor ring laser (SRL) as a novel device has received much attention, for its special cavity allows the output light to propagate in two opposite directions, namely the clockwise mode and counterclockwise mode. SRL does not require gratings or cleaved facets for optical feedback and can be a candidate for small sized photonic integrated circuits which have been developed for secure data transmission, with chaotic carriers and high rate random bit generated. In this paper, we propose a method to obtain two broadband chaotic signals with high unpredictability degree by utilizing injected slave SRL and further explore the physical mechanism and injection conditions. Based on a conventional master-slave configuration, the proposed method obtains two modes of chaotic signals by master SRL with external cross feedback, which are injected in parallel to a slave SRL correspondingly. According to the well-known Lang-Kobayashi rate equations, we establish rate equations and numerically investigate the influences of frequency detuning and injection strength on bandwidth and unpredictability degree. We adapt the given definition of bandwidth and the normalized permutation entropy to respectively evaluate bandwidth and unpredictability degree of chaotic signals. Furthermore, we reveal the underlying physical mechanism of bandwidth enhancement and asymmetric bandwidth-enhancing region by analyzing the radiofrequency and optical spectra of intensity time series. The results show that two chaotic signals have similar routes to enhancing the bandwidth in frequency domain. In the unlocking injection area, two broadband and unpredictability-enhancing chaotic signals generated by slave SRL are simultaneously achieved by choosing appropriate control parameters. Analyses of optical spectra reveal that high-frequency periodic oscillation generated between injection chaotic signals and slave light via beating is the physical mechanism of bandwidth enhncment. The bandwidthenhancing domains of two chaotic signals are asymmetrical due to redshift of master SRL frequency, with external chaotic signals injected. Bandwidth-enhanced chaotic signals are easier to obtain in the domain of negative frequency detuning. The asymmetrical injections contribute to reducing the locking region and extending the bandwidthenhancing region under high injection strength. This conventional master-slave configuration composed of two SRLs can be easily implemented on chip and save other optical devices. The slave SRL subjected to parallel injection signals from master SRL can be used as a wideband unpredictability-enhancing chaotic source, which is extremely useful for the high capacity security-enhancing multiple chaotic communications, as well as for the potential applications of high speed random number generators.