In modern mathematical systems theory, there exist two consistent ways of de. ning and describing a linear system: (i) in the behavioral approach, a linear system is the kernel B of a matrix-valued operator R in a power of a signal space W; (ii) in the module-theoretic setting, a linear system is the cokernel M of the above matrix R. These two formulations have connections. The minimal conditions under which they are equivalent are investigated in this paper. The general theory is applied to differential-difference systems over Lie groups.