G-Prime and G-Primary G-Ideals on G-Schemes

Let G be a flat finite-type group scheme over a scheme S, and X a noetherian S-scheme on which G acts. We define and study G-prime and G-primary G-ideals on X and study their basic properties. In particular, we prove the existence of minimal G-primary decomposition and the well-definedness of G-associated G-prime G-ideals. We also prove a generalization of Matijevic-Roberts type theorem. In particular, we prove Matijevic-Roberts type theorem on graded rings for F-regular and F-rational properties.