An immersion of a system is a mapping of the initial state exactly preserving the input-output map. As has already been shown, a system is immersible into a rational-in-the-state representation (RSR) and a polynomial-in-the-state representation (PSR) if and only if the field generated by the observation space is finitely generated, which is true in most practical systems. In this paper, some algebraic structures and their geometric counterparts associated with an RSR and a PSR obtained via an immersion are discussed. First, RSRs and PSRs are viewed as systems over rings in a unified framework, and the notions of an invariant ideal and an invariant variety, which are related to a differential algebraic equation, are introduced. Then, it is shown that an RSR and a PSR have invariant ideals and invariant varieties associated with an immersion. In particular, an invariant variety of an RSR or a PSR is the Zariski closure of the image of the immersion, i.e., the smallest variety containing the image of the immersion. The degrees of freedom in RSRs and PSRs obtained via immersion are also investigated and are characterized in terms of invariant ideals.