The Earth-Moon-Mars Radiation Environment Module (EMMREM) is a numerical model for characterizing the time-dependent radiation environment in the Earth-Moon-Mars and interplanetary space environments. In this work we demonstrate the capabilities of the module for performing analyses of time-dependent exposures from solar energetic particle (SEP) events near Earth and Mars by calculating time-dependent dose rates, dose equivalent rates, and accumulated dose and accumulated dose equivalents for surrogates of the skin and the blood forming organs (BFOs) of crew members shielded by as much as 10 g/cm2 of aluminum shielding for the January 15, 2005, SEP event. The motivation for the development of EMMREM is the need to better understand the radiation hazards in deep space and near Earth and other planetary bodies, in near real time in support of possible future space exploration by manned and unmanned spacecraft. Characterizing the radiation environment for different locations on and close to Earth for SEP events is fairly well developed. However, estimating the probable radiation environment near Mars and other locations throughout the solar system is not currently supported for SEP events. Such capability is critical for future human exploration of the Moon and Mars in the upcoming decades. The calculated doses for the skin and BFO surrogates are compared with the National Aeronautics and Space Administration's short-term permissible exposure limits.