The "compositeness" or "elementarity" is investigated for s-wave composite states dynamically generated by energy-dependent and independent interactions. The bare mass of the corresponding fictitious elementary particle in an equivalent Yukawa model is shown to be infinite, indicating that the wave function renormalization constant Z is equal to zero. The idea can be equally applied to both resonant and bound states. In a special case of zero-energy bound states, the condition Z = 0 does not necessarily mean that the elementary particle has the infinite bare mass. We also emphasize arbitrariness in the "elementarity" leading to multiple interpretations of a physical state, which can be either a pure composite state with Z = 0 or an elementary particle with Z not equal 0. The arbitrariness is unavoidable because the renormalization constant Z is not a physical observable.